Application of Derivatives · Mathematics · MHT CET

The area of the triangle formed by the co-ordinate axes and a tangent to the curve $x y=\mathrm{a}^2$ at the point $\left(x_1, y_1\right)$ is _______ sq. units (where a, $x_1$ and $y_1$ are non-zero)

The minimum value of the slope of the tangent to curve $y=x^3-3 x^2+2 x+93$ is

The approximate value of $\frac{1}{(2.002)^2}$ is

A spherical balloon is filled with $4500 \pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72 \pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage has begun, is

If $x$ and $y$ are sides of two squares such that $y=x-x^2$, then the rate of change of area of the second square with respect to that of the first square is

The function $\mathrm{f}(x)=[x(x-2)]^2$ is increasing in the set

The minimum value of $a x+b y$ where $x y=c^2$ is

The equation of the tangent to the curve $y=\mathrm{be}^{-x / \mathrm{a}}$ at the point where it crosses the Y axis is

A manufacturer sells $x$ items at a price of rupees $\left(6-\frac{x}{40}\right)$ each. The cost price of $x$ items is ₹ $\left(\frac{x}{5}+193\right)$. The maximum profit in ₹ __________ is

The equation of the tangent to the curve $\left(1+x^2\right) y=2-x$, where it crosses the X -axis, is

If $y=\alpha \log x+\beta x^3-x$ has extreme values at $x=-1$ and $x=1$, then $\alpha$ and $\beta$ are respectively

The length of the perpendicular drawn from the origin on the normal to the curve $x^2+2 x y-3 y^2=0$ at the point $(2,2)$ is

If $\mathrm{f}(x)=\log (1+x)-\frac{2 x}{2+x}$ then $\mathrm{f}(x)$ is increasing in

The angle $\theta$, at which the curves $y=3^x$ and $y=7^x$ intersect, is given by

The function $\mathrm{f}(x)=x^3-6 x^2+\mathrm{ax}+\mathrm{b}$ satisfies the conditions of Rolle's theorem in $[1,3]$. Then the values of $a$ and $b$ are respectively

$\mathrm{f}(x)=\frac{x}{2}+\frac{2}{x}, x \neq 0$ is strictly decreasing in

The rate of change of the volume of a sphere with respect to its surface area, when the radius is 5 m is

If the line $a x+b y+c=0$ is normal to the curve $x y=1$, then

The sum of two nonzero numbers is 4 . The minimum value of the sum of their reciprocals is

The combined equation of the tangent and normal to the curve $x y=15$ at the point $(5,3)$ is________

The length and breadth of a rectangle are $x_{x \mathrm{~cm}}$ and $y \mathrm{~cm}$ respectively. If the length decreases at the rate of $5 \mathrm{~cm} /$ minute and the breadth increases at the rate of $3 \mathrm{~cm} /$ minute, then the rates of change of the perimeter and area respectively when the length is 5 cm and breadth is 2 cm , are

The maximum value of $x^{2 / 3}+(x-2)^{2 / 3}$ is

The point on the curve $4 y^2-4 y+2 x-1=0$ at which the tangent becomes parallel to Y -axis is

A particle moves along a curve $y=\frac{2 x^3-1}{3}$. The points on the curve at which the $y$ co-ordinate is changing 18 times the $x$ co-ordinate are

The equation of motion of the particle is $\mathrm{s}=\mathrm{at}^2+\mathrm{bt}+\mathrm{c}$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} /$ seconds and the acceleration is $10 \mathrm{~m} /$ seconds $^2$, then

Let $f$ be a function which is continuous and differentiable for all $x$. If $\mathrm{f}(1)=1$ and $\mathrm{f}^{\prime}(x) \leq 5$ for all $x$ in $[1,5]$, then the maximum value of $\mathrm{f}(5)$ is

The function $\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$ increases if

The normal to the curve $x=9(1+\cos \theta)$, $y=9 \sin \theta$ at $\theta$ always passes through the fixed point

An open tank with a square bottom is to contain 4000 cubic cm . of liquid. The dimensions of the tank so that the surface area of the tank is minimum, is

If $x$ is real, then the difference between the greatest and least values of $\frac{x^2-x+1}{x^2+x+1}$ is

If $\mathrm{f}(x)=x \cdot \mathrm{e}^{x(1-x)}$, then $\mathrm{f}(x)$ is

The approximate value of $\sqrt[3]{64 \cdot 04}$ is

If $\mathrm{f}(x)=\frac{\mathrm{k} \sin x+2 \cos x}{\sin x+\cos x}$ is strictly increasing for all real values of $x$, then

The abscissae of the points of the curve $y=x^3$ are in the interval $[-2,2]$, where the slope of the tangents can be obtained by mean value theorem for the interval $[-2,2]$ are

Let $x$ be the length of each of the equal sides of an isosceles triangle and $\theta$ be the angle between these sides. If $x$ is increasing at the rate $\frac{1}{12} \mathrm{~m} /$ hour and $\theta$ is increasing at the rate $\frac{\pi}{180} \mathrm{rad} /$ hour, then the rate at which area of the triangle is increasing when $x=12 \mathrm{~m}$ and $\theta=\frac{\pi}{4}$ is

A wire of length 8 units is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

The radius of the base of a cone is increasing at the rate $3 \mathrm{~cm} /$ minute and the altitude is decreasing at the rate $4 \mathrm{~cm} /$ minute . The rate at which the lateral surface area is changing, when the radius is 7 cm and altitude is 24 cm is

The function f defined by $\mathrm{f}(x)=(x+2) \mathrm{e}^{-x}$ is

If the function $\mathrm{f}(x)=x(x+3) \mathrm{e}^{-\frac{x}{2}}$ satisfies all the conditions of Rolle's theorem in $[-3,0]$, then c is

A manufacturer produces $x$ items per week at a total cost of ₹ $\left(x^2+78 x+2500\right)$. The price per unit is given by $8 x=600-\mathrm{p}$ where ' p ' is the price of each unit. Then the maximum profit obtained is

If $2 \mathrm{f}(x)+3 \mathrm{f}\left(\frac{1}{x}\right)=x^2+1, x \neq 0$ and $y=5 x^2 \mathrm{f}(x)$, then $y$ is strictly increasing in

If the curve $y=a x^2-6 x+b$ passes through $(0,4)$ and has its tangent parallel to the X-axis at $x=\frac{3}{2}$, then the values of $a$ and $b$ respectively are

20 is divided into two parts so that the product of the cube of one part and the square of the other part is maximum, then these two parts are

The shortest distance between the line $y-x=1$ and the curve $x=y^2$ is

If the curves $y^2=6 x$ and $9 x^2+b y^2=16$ intersect each other at right angles, then the value of $b$ is

The position of a point in time $t$ is given by $x=\mathrm{a}+\mathrm{bt}-\mathrm{ct}^2, y=\mathrm{at}+\mathrm{bt}^2$. It's resultant acceleration at time $t$ in seconds is given by

The equation of tangent to the curve $y=\cos (x+y)$ where $-2 \pi \leq x \leq 2 \pi$ and which is parallel to the line $x+2 y=0$, is

If two curves $x^2-4 y^2=2$ and $8 x^2=40-\mathrm{m} y^2$ are orthogonal to each other then $\mathrm{m}=$

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is _________ $\mathrm{cm}^3 / \mathrm{cm}^2$.

Water is being poured at the rate of $36 \mathrm{~m}^3 / \mathrm{min}$ into a cylindrical vessel, whose circular base is of radius 3 meters. Then the water level in the cylinder is rising at the rate of

The equation of the normal to the curve $y=x \log x$ parallel to $2 x-2 y+3=0$ is

If $x=-1$ and $x=2$ are extreme points of $f(x)=\alpha \log |x|+\beta x^2+x$, then

If $\mathrm{f}(1)=1, \mathrm{f}^{\prime}(1)=3$, then the derivative of $\mathrm{f}(\mathrm{f}(\mathrm{f}(x)))+(\mathrm{f}(x))^2$ at $x=1$ is

If $\theta$ denotes the acute angle between the curves $y=10-x^2$ and $y=2+x^2$, at a point of the intersection, then $|\tan \theta|$ is equal to

If $y=a \log x+b x^2+x$ has its extremum values at $x=-1$ and $x=2$, then

The set of all points, for which $f(x)=x^2 e^{-x}$ strictly increases, is

The abscissa of the point on the curve $y=\mathrm{a}\left(\mathrm{e}^{\frac{x}{a}}+\mathrm{e}^{-\frac{x}{a}}\right)$ where the tangent is parallel to the X -axis is

The Number of values of C that satisfy the conclusion of Rolle's theorem in case of following function $\mathrm{f}(x)=\sin 2 \pi x, x \in[-1,1]$ is

An open tank with a square bottom, to contain 4000 cubic cm . of liquid, is to be constructed. The dimensions of the tank, so that the surface area of the tank is minimum, are

The function $\mathrm{f}(x)=2 x^3-9 x^2+12 x+2$ is decreasing in

The equation of motion of a particle is $s=a t^2+b t+c$. If the displacement after 1 second is 20 m , velocity after 2 seconds is $30 \mathrm{~m} / \mathrm{sec}$ and the acceleration is $10 \mathrm{~m} / \mathrm{sec}^2$, then

After $t$ seconds, the acceleration of a particle, which starts from rest and moves in a straight line is $\left(8-\frac{\mathrm{t}}{5}\right) \mathrm{cm} / \mathrm{s}^2$, then velocity of the particle at the instant, when the acceleration is zero, is

If $\mathrm{f}(x)=x^3-10 x^2+200 x-10$, then

Twenty meters of wire is available for fencing off a flower-bed in the form of a circular sector. Then the maximum area (in sq.m) of the flowerbed is

A ladder 5 m in length is leaning against a wall. The bottom of the ladder is pulled along the ground away from the wall, at the rate of $2 \mathrm{~m} / \mathrm{sec}$. How fast is the height on the wall decreasing when the foot of the ladder is 4 m away from the wall?

If $y=\mathrm{a} \log x+\mathrm{b} x^2+x$ has its extreme values at $x=-1$ and $x=2$, then the value of $\left(\frac{a}{b}+\frac{b}{a}\right)$ is

The curve $y=a x^3+b x^2+c x+5$ touches the X - axis at $(-2,0)$ and cuts the Y -axis at a point Q where its gradient is 3 , then values of $a, b, c$ respectively, are

The maximum value of the function $\mathrm{f}(\mathrm{x})=2 \mathrm{x}^3-15 x^2+36 x-48$ on the set $A=\left\{x / x^2+20 \leq 9 x\right\}$ is

If the normal to the curve $y=\mathrm{f}(x)$ at the point $(3,4)$ makes an angle of $\left(\frac{3 \pi}{4}\right)$ with the positive $X$-axis, then the value of $f^{\prime}(3)$ is

If $\mathrm{f}(x)=x^3-6 x^2+9 x+3$ is monotonically decreasing function, then $x$ lies in

If equation of normal to the curve $x=\sqrt{t}$, $y=\mathrm{t}-\frac{1}{\sqrt{\mathrm{t}}}$ at $\mathrm{t}=4$ is

The rate of change of the volume of a sphere with respect to its surface area, when its radius is 2 cm , is

The distance ' $s$ ' in meters covered by a body in $t$ seconds is given by $s=3 t^2-8 t+5$. The body will stop after

The volume of a ball is increasing at the rate of $4 \pi \mathrm{cc} / \mathrm{sec}$. The rate of increase of the radius, when the volume is $288 \pi \mathrm{cc}$, is

Let $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$. Values of C will be __________ if L.M.V.T. (Lagrange's Mean Value Theorem) can be applied.

If $y=4 x-5$ is a tangent to the curve $y^2=p x^3+q$ at $(2,3)$, then the values of $p$ and $q$ are respectively

Water is running in a hemispherical bowl of radius 180 cm at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is 120 cm ? ( 1 decimeter $=10 \mathrm{~cm}$)

A point moves along the arc of parabola $y=2 x^2$. Its abscissa increases uniformly at the rate of 2 units $/ \mathrm{sec}$. At the instant, the point is passing through ( 1,2 ), its distance from origin is increasing at the rate of

The equation of the normal to the curve $y=x \log x$, which is parallel to the line $2 x-2 y+3=0$, is

The function $\mathrm{f}(x)=2 x^3-6 x+5$ is an increasing function, if

A square plate is contracting at the uniform rate $3 \mathrm{~cm}^2 / \mathrm{sec}$, then the rate at which the perimeter is decreasing, when the side of the square is 15 cm , is

A poster is to be printed on a rectangular sheet of paper of area $18 \mathrm{~m}^2$. The margins at the top and bottom of 75 cm each and at the sides 50 cm each are to be left. Then the dimensions i.e. height and breadth of the sheet so that the space available for printing is maximum, are _______ respectively.

The equation of the tangent to the curve $x=\operatorname{acos}^3 \theta, y=\operatorname{asin}^3 \theta$ at $\theta=\frac{\pi}{4}$ is

Let C be a curve given by $y(x)=1+\sqrt{4 x-3}$, $x>\frac{3}{4}$. If P is a point on C , such that the tangent at P has slope $\frac{2}{3}$, then a point through which the normal at P passes, is

If $\mathrm{f}(x)=\frac{\log x}{x}(x>0)$, then it is increasing in

The maximum value of $\frac{\log x}{x}$ is

If the curves $y^2=6 x, 9 x^2+\mathrm{b} y^2=16$ intersect each other at right angles, then the value of $b$ is

If Mean value theorem holds for the function $\mathrm{f}(x)=(x-1)(x-2)(x-3), x \in[0,4]$ then the values of $c$ as per the theorem are

A ladder 5 m long rests against a vertical wall. If its top slides downwards at the rate of $10 \mathrm{~cm} / \mathrm{sec}$., then the foot of the ladder is sliding at the rate of _________ $\mathrm{m} / \mathrm{sec}$., when it is 4 m away from the wall.

The equation of the tangent to the curve $y=1-\mathrm{e}^{\frac{x}{3}}$ at the point of intersection with Y -axis is

If $8 \mathrm{f}(x)+6 \mathrm{f}\left(\frac{1}{x}\right)=x+5$ and $y=x^2 \mathrm{f}(x)$, then $\frac{\mathrm{d} y}{\mathrm{~d} x}$ at $x=-1$ is

The function $f(x)=\frac{\log _e(\pi+x)}{\log _e(e+x)}$ is

The sum of intercepts on coordinate axes made by tangent to the curve $\sqrt{x}+\sqrt{y}=\sqrt{a}$ is

A wire of length 2 units is cut into two parts, which are bent respectively to form a square of side $x$ units and a circle of radius of r units. If the sum of the areas of square and the circle so formed is minimum, then

If sum of two numbers is 3 , then the maximum value of the product of first number and square of the second number is

The curve $y=a x^3+b x^2+c x+5$ touches the $x$-axis at $(-2,0)$ and cuts the $y$-axis at a point Q where its gradient is 3 , then the value of $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is

If $y=a \log x+b x^2+x$ has its extreme value at $x=-1$ and $x=2$, then the value of $a+b$ is

If $\mathrm{f}(x)=x^3+b x^2+c x+d$ and $0< b^2< c$, then in $(-\infty, \infty)$

The minimum value of the function $\mathrm{f}(x)=2 x^3-15 x^2+36 x-48$ on the set $\mathrm{A}=\left\{x \mid x^2+20 \leqslant 9 x\right\}$ is

The value of c for which Rolle's theorem for the function $\mathrm{f}(x)=x^3-3 x^2+2 x$ in the interval $[0,2]$ are

A triangular park is enclosed on two sides by a fence and on the third side a straight river bank. The two sides having fence are of same length $x$. The maximum area (in sq. units) enclosed by the park is

A stone is dropped into a quiet lake and waves move in circles at speed of $8 \mathrm{~cm} / \mathrm{sec}$. At the instant when the radius of the circular wave is 12 cm . how fast is the enclosed area increasing?

If the half life of substance is 5 years, then the total amount of the substance left after 15 years, when initial amount is 64 gms is

A bullet is shot horizontally and its distance S cm at time t second is given by $\mathrm{S}=1200 \mathrm{t}-15 \mathrm{t}^2$, then the distance covered by the bullet when it comes to the rest, is

The equation of the normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

The co-ordinates of a point on the curve $y=x \log x$ at which the normal is parallel to the line $2 x-2 y=3$ are

The value of C for which Mean value Theorem holds for the function $\mathrm{f}(x)=\log _e x$ on the interval $[1,3]$ is

The maximum value of the function

$$f(x)=3 x^3-18 x^2+27 x-40$$

on the set $\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$ is

The equation of normal to the curve $x=\theta+\sin \theta, y=1+\cos \theta$ at $\theta=\frac{\pi}{2}$ is

The approximate value of $\sqrt[3]{0.026}$ is

The length of the longest interval, in which the function $3 \sin x-4 \sin ^3 x$ is increasing, is

If Rolle's theorem holds for the function $\mathrm{f}(x)=x^3+\mathrm{bx}{ }^2+\mathrm{ax}+5$ on $[1,3]$ with $\mathrm{c}=2+\frac{1}{\sqrt{3}}$, then the values of $a$ and $b$ respectively are

The normal to the curve, $y(x-2)(x-3)=x+6$ at the point, where the curve intersects the Y-axis, passes through the point

The function $$\mathrm{f}(x)=x^3-6 x^2+9 x+2$$ has maximum value when $$x$$ is

If $$y=4 x-5$$ is a tangent to the curve $$y^2=\mathrm{p} x^3+\mathrm{q}$$ at $$(2,3)$$, then $$\mathrm{p}-\mathrm{q}$$ is

The diagonal of a square is changing at the rate of $$0.5 \mathrm{~cm} / \mathrm{sec}$$. Then the rate of change of area when the area is $$400 \mathrm{~cm}^2$$ is equal to

Let $$x_0$$ be the point of local minima of $$\mathrm{f}(x)=\overline{\mathrm{a}} \cdot(\overline{\mathrm{b}} \times \overline{\mathrm{c}})$$ where $$\overline{\mathrm{a}}=x \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+3 \hat{\mathrm{k}}, \overline{\mathrm{b}}=-2 \hat{\mathrm{i}}+x \hat{\mathrm{j}}-\hat{\mathrm{k}}, \overline{\mathrm{c}}=7 \hat{\mathrm{i}}-2 \hat{\mathrm{j}}+x \hat{\mathrm{k}}$$, then value of $$\overline{\mathrm{a}} \cdot \overline{\mathrm{b}}$$ at $$x=x_0$$ is

Let the curve be represented by $$x=2(\cos t+t \sin t), y=2(\sin t-t \cos t)$$. Then normal at any point '$$t$$' of the curve is at a distance of ______ units from the origin.

Let $$\mathrm{B} \equiv(0,3)$$ and $$\mathrm{C} \equiv(4,0)$$. The point $$\mathrm{A}$$ is moving on the line $$y=2 x$$ at the rate of 2 units/second. The area of $$\triangle \mathrm{ABC}$$ is increasing at the rate of

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathrm{R} / x^2+30 \leq 11 x\right\}$$ is

Slope of the tangent to the curve $$y=2 e^x \sin \left(\frac{\pi}{4}-\frac{x}{2}\right) \cos \left(\frac{\pi}{4}-\frac{x}{2}\right)$$, where $$0 \leq x \leq 2 \pi$$ is minimum at $$x=$$

If slope of the tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then the value of $$3 a+b$$ is

$$A(1,-3), B(4,3)$$ are two points on the curve $$y=x-\frac{4}{x}$$. The points on the curve, the tangents at which are parallel to the chord $$A B$$, are

Water is running in a hemispherical bowl of radius $$180 \mathrm{~cm}$$ at the rate of 108 cubic decimeters per minute. How fast the water level is rising when depth of the water level in the bowl is $$120 \mathrm{~cm}$$ ? (1 decimeter $$=10 \mathrm{~m}$$)

If Rolle's theorem holds for the function $$f(x)=x^3+b x^2+a x+5$$ on $$[1,3]$$ with $$c=2+\frac{1}{\sqrt{3}}$$, then the values of $$a$$ and $$b$$ respectively are

If $$\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$$ and $$0<\mathrm{b}^2<\mathrm{c}$$, then in $$(-\infty, \infty)$$

The slope of the normal to the curve $$x=\sqrt{t}$$ and $$y=t-\frac{1}{\sqrt{t}}$$ at $$t=4$$ is

Values of $$c$$ as per Rolle's theorem for $$f(x)=\sin x+\cos x+6$$ on $$[0,2 \pi]$$ are

A poster is to be printed on a rectangular sheet of paper of area $$18 \mathrm{~m}^2$$. The margins at the top and bottom of $$75 \mathrm{~cm}$$ each and at the sides $$50 \mathrm{~cm}$$ each are to be left. Then the dimensions i.e. height and breadth of the sheet, so that the space available for printing is maximum, are ________ respectively.

The equation of the normal to the curve $$3 x^2-y^2=8$$, which is parallel to the line $$x+3 y=10$$, is

If the curves $$y^2=6 x$$ and $$9 x^2+b y^2=16$$ intersect each other at right angle, then value of '$$b$$' is

A ladder, 5 meters long, rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle between the ladder and the floor is decreasing at the rate of __________ radians/second when it's lower end is $$4 \mathrm{~m}$$ away from the wall.

A tank with a rectangular base and rectangular sides, open at the top is to be constructed so that its depth is 4 meter and volume is 36 cubic meters. If building of the tank costs ₹ 100 per square meter for the base and ₹ 50 per square meter for the sides, then the cost of least expensive tank is

The angle between the tangents to the curves $$y=2 x^2$$ and $$x=2 y^2$$ at $$(1,1)$$ is

The function $$\mathrm{f}(x)=\sin ^4 x+\cos ^4 x$$ is increasing in

A ladder 5 meters long rests against a vertical wall. If its top slides downwards at the rate of $$10 \mathrm{~cm} / \mathrm{s}$$, then the angle between the ladder and the floor is decreasing at the rate of ________ rad./s when it's lower end is $$4 \mathrm{~m}$$ away from the wall.

If the function $$f$$ is given by $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$\mathrm{a} \in \mathbb{R}$$, is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equation $$\frac{\mathrm{f}(x)-14}{(x-1)^2}=0(x \neq 1)$$ is

If $$a$$ and $$b$$ are positive number such that $$a>b$$, then the minimum value of $$a \sec \theta-b \tan \theta\left(0 < \theta < \frac{\pi}{2}\right)$$ is

$$A$$ rod $$A B, 13$$ feet long moves with its ends $$A$$ and $$B$$ on two perpendicular lines $$O X$$ and $$O Y$$ respectively. When $$A$$ is 5 feet from $$O$$, it is moving away at the rate of $$3 \mathrm{feet} / \mathrm{sec}$$. At this instant, $$\mathrm{B}$$ is moving at the rate

The equation of the tangent to the curve $$y=\sqrt{9-2 x^2}$$, at the point where the ordinate and abscissa are equal, is

At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production $$\mathrm{P}$$ w.r.t. additional number of worker $$x$$ is given by $$\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}$$. If the firm employees 9 more workers, then the new level of production of items is

Value of $$c$$ satisfying the conditions and conclusions of Rolle's theorem for the function $$\mathrm{f}(x)=x \sqrt{x+6}, x \in[-6,0]$$ is

If $$\mathrm{f}(x)=x \mathrm{e}^{x(1-x)}, x \in \mathrm{R}$$, then $$\mathrm{f}(x)$$ is

The value of $$\mathrm{c}$$ for the function $$\mathrm{f}(x)=\log x$$ on [$$1$$, e] if LMVT can be applied, is

Let $$\mathrm{f}(x)=\mathrm{e}^x-x$$ and $$\mathrm{g}(x)=x^2-x, \forall x \in \mathrm{R}$$, then the set of all $$x \in \mathrm{R}$$, where the function $$\mathrm{h}(x)=(\mathrm{fog})(x)$$ is increasing is

The displacement '$$\mathrm{S}$$' of a moving particle at a time $$t$$ is given by $$S=5+\frac{48}{t}+t^3$$. Then its acceleration when the velocity is zero, is

If the surface area of a spherical balloon of radius $$6 \mathrm{~cm}$$ is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate of increase in its volume in $$\mathrm{cm}^3 / \mathrm{sec}$$ is

The value of $$\alpha$$, so that the volume of parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{\mathrm{i}}+\hat{\mathrm{k}}$$ becomes minimum, is

In a certain culture of bacteria, the rate of increase is proportional to the number of bacteria present at that instant. It is found that there are 10,000 bacteria at the end of 3 hours and 40,000 bacteria at the end of 5 hours, then the number of bacteria present in the beginning are

An open metallic tank is to be constructed, with a square base and vertical sides, having volume 500 cubic meter. Then the dimensions of the tank, for minimum area of the sheet metal used in its construction, are

A square plate is contracting at the uniform rate $$4 \mathrm{~cm}^2 / \mathrm{sec}$$, then the rate at which the perimeter is decreasing, when side of the square is $$20 \mathrm{~cm}$$, is

A ladder of length $$17 \mathrm{~m}$$ rests with one end against a vertical wall and the other on the level ground. If the lower end slips away at the rate of $$1 \mathrm{~m} / \mathrm{sec}$$., then when it is $$8 \mathrm{~m}$$ away from the wall, its upper end is coming down at the rate of

If the line $$a x+b y+c=0$$ is a normal to the curve $$x y=1$$, then

A kite is $$120 \mathrm{~m}$$ high and $$130 \mathrm{~m}$$ of string is out. If the kite is moving away horizontally at the rate of $$39 \mathrm{~m} / \mathrm{sec}$$, then the rate at which the string is being out, is

If slope of a tangent to the curve $$x y+a x+b y=0$$ at the point $$(1,1)$$ on it is 2, then a - b is

The equation $$x^3+x-1=0$$ has

The range of values of $$x$$ for which $$f(x)=x^3+6 x^2-36 x+7$$ is increasing in

The maximum value of the function $$f(x)=3 x^3-18 x^2+27 x-40$$ on the set $$\mathrm{S}=\left\{x \in \mathbb{R} / x^2+30 \leq 11 x\right\}$$ is

A water tank has a shape of inverted right circular cone whose semi-vertical angle is $$\tan ^{-1}\left(\frac{1}{2}\right)$$. Water is poured into it at constant rate of 5 cubic meter/minute. The rate in meter/ minute at which level of water is rising, at the instant when depth of water in the tank is $$10 \mathrm{~m}$$ is

Let $$\mathrm{f}(0)=-3$$ and $$\mathrm{f}^{\prime}(x) \leq 5$$ for all real values of $$x$$. The $$\mathrm{f}(2)$$ can have possible maximum value as

The value of $$c$$ of Lagrange's mean value theorem for $$f(x)=\sqrt{25-x^2}$$ on $$[1,5]$$ is

The value of $$\alpha$$, so that the volume of the parallelopiped formed by $$\hat{i}+\alpha \hat{j}+\hat{k}, \hat{j}+\alpha \hat{k}$$ and $$\alpha \hat{i}+\hat{k}$$ becomes maximum, is

The maximum value of xy when x + 2y = 8 is

An object is moving in the clockwise direction around the unit circle $$x^2+y^2=1$$. As it passes through the point $$\left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$, its $$y$$-co-ordinate is decreasing at the rate of 3 units per sec. The rate at which the $$x$$-co-ordinate changes at this point is

A spherical raindrop evaporates at a rate proportional to its surface area. If originally its radius is $$3 \mathrm{~mm}$$ and 1 hour later it reduces to $$2 \mathrm{~mm}$$, then the expression for the radius $$R$$ of the raindrop at any time $$t$$ is

If the function $$f(x)=x^3-3(a-2) x^2+3 a x+7$$, for some $$a \in I R$$ is increasing in $$(0,1]$$ and decreasing in $$[1,5)$$, then a root of the equation $$\frac{f(x)-14}{(x-1)^2}=0(x \neq 1)$$ is

A firm is manufacturing 2000 items. It is estimated that the rate of change of production $$P$$ with respect to additional number of workers $$x$$ is given by $$\frac{\mathrm{d} P}{\mathrm{~d} x}=100-12 \sqrt{x}$$. If the firm employs 25 more workers, then the new level of production of items is

If the normal to the curve $$y=f(x)$$ at the point $$(3,4)$$ makes an angle $$\left(\frac{3 \pi}{4}\right)^c$$ with positive $$X$$-axis, then $$f^{\prime}(3)$$ is equal to

If $$y=\cos \left(\sin x^2\right)$$, then $$\frac{\mathrm{d} y}{\mathrm{~d} x}$$ at $$x=\sqrt{\frac{\pi}{2}}$$ is

A spherical iron ball of $$10 \mathrm{~cm}$$ radius is coated with a layer of ice of uniform thickness that melts at the rate of $$50 \mathrm{~cm}^3 / \mathrm{min}$$. If the thickness of ice is $$5 \mathrm{~cm}$$, then the rate at which the thickness of ice decreases is

The distance 's' in meters covered by a particle in t seconds is given by $$s=2+27 t-t^3$$. The particle will stop after _________ distance.

The curve $$y=a x^3+b x^2+c x+5$$ touches $$X$$-axis at $$P(-2,0)$$ and cuts $$Y$$-axis at a point $$Q$$, where its gradient is 3, then

The minimum value of the function f(x) = x log x is

The maximum area of the rectangle that can be inscribed in a circle of radius $$r$$ is

$$f(x)=\log |\sin x|$$, where $$x \in(0, \pi)$$ is strictly increasing on

The velocity of a particle at time $$t$$ is given by the relation $$v=6 t-\frac{t^2}{6}$$. Its displacement S is zero at $$\mathrm{t}=0$$, then the distance travelled in $$3 \mathrm{~sec}$$ is

The radius of a circular plate is increasing at the rate of $$0.01 \mathrm{~cm} / \mathrm{sec}$$, when the radius is $$12 \mathrm{~cm}$$. Then the rate at which the area increases is

If $$x=a(\theta+\sin \theta)$$ and $$y=a(1-\cos \theta)$$ then $$\left(\frac{d^2 y}{d x^2}\right)_{at~ \theta=\pi / 2}=$$

The equation of tangent to the curve $$y=\sqrt{2} \sin \left(2 x+\frac{\pi}{4}\right)$$ at $$x=\frac{\pi}{4}$$, is

Function $$f(x)=e^{-1 / x}$$ is strictly increasing for all $$x$$ where

If $$x=-2$$ and $$x=4$$ are the extreme points of $$y=x^3-\alpha x^2-\beta x+5$$, then

10 is divided into two parts such that the sum of double of the first and square of the other is minimum, then the numbers are respectively

The function $$f(x)=\frac{\lambda \sin x+6 \cos x}{2 \sin x+3 \cos x}$$ is increasing, if

If $$f(x)=x^2+a x+b$$ has minima at $$x=3$$ whose value is 5 , then the values of $$a$$ and $$b$$ are respectively.

The slant height of a right circular cone is $$3 \mathrm{~cm}$$. The height of the cone for maximum volume is

The point on the curve $$y^2=2(x-3)$$ at which the normal is parallel to the line $$y-2 x+1=0$$ is

A sperical snow ball is forming so that its volume is increasing at the rate of $$8 \mathrm{~cm}^3 / \mathrm{sec}$$. Find the rate of increase of radius when radius is $$2 \mathrm{~cm}$$.

The abscissa of the points, where the tangent to the curve $$y=x^3-3 x^2-9 x+5$$ is parallel to $$X$$ axis are

For all real $$x$$, the minimum value of the function $$f(x)=\frac{1-x+x^2}{1+x+x^2}$$ is

The function $$f(x)=\log (1+x)-\frac{2 x}{2+x}$$ is increasing on

A body at an unknown temperature is placed in a room which is held at a constant temperature of $$30^{\circ} \mathrm{F}$$. If after 10 minutes the temperature of the body is $$0^{\circ} \mathrm{F}$$ and after 20 minutes the temperature of the body is $$15^{\circ} \mathrm{F}$$, then the expression for the temperature of the body at any time $$\mathrm{t}$$ is

A stone is thrown into a quite lake and the waves formed move in circles. If the radius of a circular wave increases at the rate of 4 cm/sec, then the rate of increase in its area, at the instant when its radius is 10 cm, is _________ cm$$^2$$/sec.

The function $$f(x)=\cot ^{-1} x+x$$ is increasing in the interval.

The curves $$\frac{x^2}{a^2}+\frac{y^2}{4}=1$$ and $$y^3=16 x$$ intersect each other orthogonally, then $$a^2=$$

The surface area of a spherical balloon is increasing at the rate $$2 \mathrm{~cm}^2 / \mathrm{sec}$$. Then rate of increase in the volume of the balloon is , when the radius of the balloon is $$6 \mathrm{~cm}$$.

If $$f(x)=2x^3-15x^2-144x-7$$, then $$f(x)$$ is strictly decreasing in

The equation of tangent to the circle $$x^2+y^2=64$$ at the point $$\mathrm{P\left(\frac{2\pi}{3}\right)}$$ is

Water is being poured at the rate of 36 m$$^3$$/min. into a cylindrical vessel, whose circular base is of radius 3 m. Then the wate level in the cylinder is rising at the rate of

A wire of length 20 units is divided into two parts such that the product of one part and cube of the other part is maximum, then product of these parts is

A particle is moving on a straight line. The distance $$\mathrm{S}$$ travelled in time $$\mathrm{t}$$ is given by $$\mathrm{S=a t^2+b t+6}$$. If the particle comes to rest after 4 seconds at a distance of $$16 \mathrm{~m}$$. from the starting point, then the acceleration of the particle is.

The equation of the tangent to the curve $$y = 4x{e^x}$$ at $$\left( { - 1,{{ - 4} \over e}} \right)$$ is

The maximum volume of a right circular cylinder if the sum of its radius and height is 6 m is

The equation of the normal to the curve $2 x^2+y^2=12$ at the point $(2,2)$ is

The area of the square increases at the rate of $0.5 \mathrm{~cm}^2 / \mathrm{sec}$. The rate at which its perimeter is increasing when the side of the square is 10 cm long, is

The equation of normal to the curve $$y=\sin \left(\frac{\pi x}{4}\right)$$ at the point $$(2,5)$$ is

$$\text { If } \sin (x+y)+\cos (x+y)=\sin \left[\cos ^{-1}\left(\frac{1}{3}\right)\right] \text {, then } \frac{dy}{dx}=$$

For every value of $$x$$, the function $$f(x)=\frac{1}{a^x}, a>$$ 0 is,

If $$f(x)=\log (\sin x), x \in\left[\frac{\pi}{6}, \frac{5 \pi}{6}\right]$$, then value of '$$c$$' by applying LMVT is

The equation of tangent at $$P(-4,-4)$$ on the curve $$x^2=-4 y$$ is

- The edge of a cube is decreasing at the rate of $0.04 \mathrm{~cm} / \mathrm{sec}$. If the edge of the cube is 10 cms , then rate of decrease of surface area of the cube is...

If $r$ is the radius of spherical balloon at time $t$ and the surface area of balloon changes at a constant rate $K$, then......

The slope of normal to the curve $x=\sqrt{t}$ and $y=t-\frac{1}{\sqrt{t}}$ at $t=4$ is ..........

If $f(x)=x+\frac{1}{x}, x \neq 0$, then local maximum and minimum values of function $f$ are respectively.......

The function $f(x)=x^3-3 x$ is ............

Using differentiation, approximate value of $f(x)=x^2-2 x+1$ at $x=2.99$ is ............

A particle moves so that $x=2+27 t-t^3$. The direction of motion reverses after moving a distance of ....... units.

A stone is dropped into a pond. Waves in the form of circles are generated and radius of outermost ripple increases at the rate of $5 \mathrm{~cm} / \mathrm{sec}$. Then area increased after 2 seconds is ............

The equation of normal to the curve $y=\log _\theta x$ at the point $P(1,0)$ is ............

If $f(x)=3 x^3-9 x^2-27 x+15$, then the maximum value of $f(x)$ is ...........