Application of Derivatives · Mathematics · COMEDK

If the function $f(x)=\mu \sin x+\frac{1}{3} \sin 3 x$ has its derivative equal to zero at $x=\frac{\pi}{3}$, then the value of ' $\mu$ ' is

A man is moving away from a tower 41.6 m high at a rate of $2 \mathrm{~m} / \mathrm{s}$. If the eyelevel of the man is 1.6 m above the ground, then the rate at which the angle of elevation of the top of the tower changes, when he is at a distance of 30 m from the foot of the tower is :

If a quadratic function in $x$ has the value 19 when $x=1$ and has a maximum value 20 when $x=2$, then the function is

If the length of the diagonal of a square is increasing at the rate of $0.1 \mathrm{~cm} / \mathrm{sec}$. What is the rate of increase of its area when the side is $\frac{15}{\sqrt{2}} \mathrm{~cm}$ ?

The function $y=\frac{\log x}{x^3}$ is strictly increasing function for

The curve $4 y=3 x^4-2 x^2$ attains ----------- at the points $x=-\frac{1}{\sqrt{3}}$ and $x=\frac{1}{\sqrt{3}}$

$x=a(\theta+\sin \theta)$ and $y=a(1-\cos \theta)$ represents the equation of a curve. If $\theta$ changes at a constant rate $k$ then the rate of change of the slope of the tangent to the curve at $\theta=\frac{\pi}{3}$ is

Quadrilateral PQRS is inscribed inside a rectangle of dimensions $10 \mathrm{~cm} \times 8 \mathrm{~cm}$. The value of ' $x$ ', if the area of the quadrilateral is minimum is

The least area of a circle circumscribing any right-angle triangle of area $\frac{9}{\pi}$ sq units is

Let $A$ and $G$ denote the arithmetic mean and geometric mean of positive real numbers $5^x$ and $5^{1-x}$. Then the minimum value of the expression $5^x+5^{1-x}$ where $x \in R$ is

In the interval $(0,1)$ the function $f(x)=x^2-x+1$ is

The least value of ' $a$ ' such that the function $x^2+a x+1$ is increasing on $[1,2]$ is

A spherical snowball is melting such that its volume is decreasing at the rate of $1 \mathrm{~cm}^3 / \mathrm{min}$. The rate at which the diameter is decreasing when the diameter is 10 cm is

The curve $a x^3+b x^2+c x+d$ has a point of minima at $x=1$, then

Oil from a conical funnel is dripping at the rate of $5 \mathrm{~cm}^3 / \mathrm{s}$. If the radius and height of the funnel are 10 cm and 20 cm respectively, then the rate at which the oil level drops when it is 5 cm from the top is

A solid S is made from a cylinder surmounted by a hemisphere on top with both its circular faces sharing a common centre. The radius of cylinder and radius of hemisphere are $x \mathrm{~cm}$. The height of the cylinder is $(20-4 x) \mathrm{cm}$ and the volume of S is $V=\frac{1}{3} \pi y$. Find the maximum value of $y$.

$$ \text { The rate of change of the volume of a sphere with respect to its surface area } \mathrm{S} \text { is } $$

The turning point of the function $$y=\frac{a x-b}{(x-1)(x-4)}$$ at the point $$P(2,-1)$$ is

The side of a cube is equal to the diameter of a sphere. If the side and radius increase at the same rate then the ratio of the increase of their surface area is

What is the nature of the function $$f(x)=x^3-3 x^2+4 x$$ on real numbers?

$$ \text { If } f(x)=\frac{a \sin x+b \cos x}{c \sin x+d \cos x} \text { is decreasing for all } x \text {, then } $$

$$ \text { The function } y=\tan x-x \text { is } $$

If $$f(x)=\log x+b x^2+a x, x \neq 0$$ has extreme values (or turning points) at $$x=-1$$ and $$x=2$$ then the values of $$\mathrm{a}$$ and $$\mathrm{b}$$ are

The dimensions of the largest rectangle of side $$x$$ and $$y$$ that can be inscribed in the right angled triangle of sides $$\mathrm{a}$$ and $$\mathrm{b}$$ is

If $$(x-a)^2+(y-b)^2=c^2$$, where $$\mathrm{a}, \mathrm{b}, \mathrm{c}$$ are some constants, $$c>0$$ then $$\frac{\left[1+\left(\frac{d y}{d x}\right)^2\right]^{\frac{3}{2}}}{\frac{d^2 y}{d x^2}}$$ is independent of

The side of an equilateral triangle expands at the rate of $$\sqrt{3} \mathrm{~cm} / \mathrm{sec}$$. When the side is $$12 \mathrm{~cm}$$, the rate of increase of its area is

If $$f(x)=2 x^3+9 x^2+\lambda x+20$$ is a decreasing function of $$x$$ in the largest possible interval $$(-2,-1)$$, then $$\lambda$$ is equal to

$$ \text { The point on the curve } x^2=x y \text { which is closest to }(0,5) \text { is } $$

For a given curve $$y=2 x-x^2$$, when $$x$$ increases at the rate of 3 units/sec, then how does the slope of the curve change?

The most economical proportion of the height of a covered box of fixed volume whose base is a rectangle with one side three times as long as the other, is

COMEDK 2023 Morning Shift

The slope of the tangent to the curve, $$y=x^2-x y$$ at $$\left(1, \frac{1}{2}\right)$$ is

Let $$f(x)=a+(x-4)^{\frac{4}{9}}$$, then minima of $$f(x)$$ is

COMEDK 2023 Morning Shift

The function $$f(x)=\frac{x}{2}+\frac{2}{x}$$ has a local minimum at

$$ f(x)=2 x-\tan ^{-1} x-\log (x+\sqrt{x^2+1}) \text { is monotonically increasing, when } $$

The altitude of a cone is $$20 \mathrm{~cm}$$ and its semi vertical angle is $$30^{\circ}$$. If the semi vertical angle is increasing at the rate of $$2^0$$ per second, then the radius of the base is increasing at the rate of

If the volume of a sphere is increasing at a constant rate, then the rate at which its radius is increasing is

If the tangent to the curve $$xy + ax + by = 0$$ at (1, 1) is inclined at an angle $${\tan ^{ - 1}}2$$ with X-axis, then

COMEDK 2022

Let $$f(x) = a - {(x - 3)^{8/9}}$$, then maxima of $$f(x)$$ is

COMEDK 2022

The approximate value of $$f(5.001)$$, where $$f(x)=x^3-7x^2+15$$ is

COMEDK 2021

Find the maximum value of $$f(x) = {1 \over {4{x^2} + 2x + 1}}$$.

COMEDK 2021

The point on the curve $$y^2=x$$, the tangent at which makes an angle 45$$^\circ$$ with X-axis is

The length of the subtangent to the curve $${x^2}{y^2} = {a^4}$$ at $$( - a,a)$$ is

The range in which $$y = - {x^2} + 6x - 3$$ is increasing, is

OA and OB are two roads enclosing an angle of 120$$^\circ$$. X and Y start from O at the same time. X travels along OA with a speed of 4 km/h and Y travels along OB with a speed of 3 km/h. The rate at which the shortest distance between X and Y is increasing after 1 hour is