Application of Derivatives · Mathematics · JEE Main

Let the set of all values of $$p$$, for which $$f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7$$ does not have any critica...

Let the set of all positive values of $$\lambda$$, for which the point of local minimum of the function $$(1+x(\lambda^2-x^2))$$ satisfies $$\frac{x^2...

Let $$\mathrm{A}$$ be the region enclosed by the parabola $$y^2=2 x$$ and the line $$x=24$$. Then the maximum area of the rectangle inscribed in the r...

Let the maximum and minimum values of $$\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}$$ be $$\mathrm{M}$$ and $$\mathrm{m}$$, respectiv...

Let $$f(x)=2^x-x^2, x \in \mathbb{R}$$. If $$m$$ and $$n$$ are respectively the number of points at which the curves $$y=f(x)$$ and $$y=f^{\prime}(x)$...

Let for a differentiable function $f:(0, \infty) \rightarrow \mathbf{R}, f(x)-f(y) \geqslant \log _{\mathrm{e}}\left(\frac{x}{y}\right)+x-y, \forall x...

Consider the triangles with vertices $A(2,1), B(0,0)$ and $C(t, 4), t \in[0,4]$. If the maximum and the minimum perimeters of such triangles are obta...

Let the quadratic curve passing through the point $$(-1,0)$$ and touching the line $$y=x$$ at $$(1,1)$$ be $$y=f(x)$$. Then the $$x$$-intercept of the...

If $$a_{\alpha}$$ is the greatest term in the sequence $$\alpha_{n}=\frac{n^{3}}{n^{4}+147}, n=1,2,3, \ldots$$, then $$\alpha$$ is equal to __________...

Let a curve $$y=f(x), x \in(0, \infty)$$ pass through the points $$P\left(1, \frac{3}{2}\right)$$ and $$Q\left(a, \frac{1}{2}\right)$$. If the tangent...

The number of points, where the curve $$y=x^{5}-20 x^{3}+50 x+2$$ crosses the $$\mathrm{x}$$-axis, is ____________.

If the equation of the normal to the curve $$y = {{x - a} \over {(x + b)(x - 2)}}$$ at the point (1, $$-$$3) is $$x - 4y = 13$$, then the value of $$a...

If the tangent to the curve $$y=x^{3}-x^{2}+x$$ at the point $$(a, b)$$ is also tangent to the curve $$y = 5{x^2} + 2x - 25$$ at the point (2, $$-$$1)...

A water tank has the shape of a right circular cone with axis vertical and vertex downwards. Its semi-vertical angle is $$\tan ^{-1} \frac{3}{4}$$. Wa...

Let $$M$$ and $$N$$ be the number of points on the curve $$y^{5}-9 x y+2 x=0$$, where the tangents to the curve are parallel to $$x$$-axis and $$y$$-a...

Let the function $$f(x)=2 x^{2}-\log _{\mathrm{e}} x, x>0$$, be decreasing in $$(0, \mathrm{a})$$ and increasing in $$(\mathrm{a}, 4)$$. A tangent to ...

The sum of the maximum and minimum values of the function $$f(x)=|5 x-7|+\left[x^{2}+2 x\right]$$ in the interval $$\left[\frac{5}{4}, 2\right]$$, whe...

A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate a...

Let l be a line which is normal to the curve y = 2x2 + x + 2 at a point P on the curve. If the point Q(6, 4) lies on the line l and O is origin, then ...

Let $$f(x) = |(x - 1)({x^2} - 2x - 3)| + x - 3,\,x \in R$$. If m and M are respectively the number of points of local minimum and local maximum of f i...

Let f(x) be a cubic polynomial with f(1) = $$-$$10, f($$-$$1) = 6, and has a local minima at x = 1, and f'(x) has a local minima at x = $$-$$1. Then f...

If 'R' is the least value of 'a' such that the function f(x) = x2 + ax + 1 is increasing on [1, 2] and 'S' is the greatest value of 'a' such that the ...

The number of distinct real roots of the equation 3x4 + 4x3 $$-$$ 12x2 + 4 = 0 is _____________.

A wire of length 36 m is cut into two pieces, one of the pieces is bent to form a square and the other is bent to form a circle. If the sum of the are...

Let f : [$$-$$1, 1] $$ \to $$ R be defined as f(x) = ax2 + bx + c for all x$$\in$$[$$-$$1, 1], where a, b, c$$\in$$R such that f($$-$$1) = 2, f'($$-$$...

Let the normals at all the points on a given curve pass through a fixed point (a, b). If the curve passes through (3, $$-$$3) and (4, $$-$$2$$\sqrt 2 ...

Let a be an integer such that all the real roots of the polynomial 2x5 + 5x4 + 10x3 + 10x2 + 10x + 10 lie in the interval (a, a + 1). Then, |a| is equ...

If the curves x = y4 and xy = k cut at right angles, then (4k)6 is equal to __________.

Let f(x) be a polynomial of degree 6 in x, in which the coefficient of x6 is unity and it has extrema at x = $$-$$1 and x = 1. If $$\mathop {\lim }\li...

The minimum value of $$\alpha $$ for which the equation $${4 \over {\sin x}} + {1 \over {1 - \sin x}} = \alpha $$ has at least one solution in $$\lef...

If the lines x + y = a and x – y = b touch the curve y = x2 – 3x + 2 at the points where the curve intersects the x-axis, then $${a \over b}$$ is equ...

Let ƒ(x) be a polynomial of degree 3 such that ƒ(–1) = 10, ƒ(1) = –6, ƒ(x) has a critical point at x = –1 and ƒ'(x) has a critical point at x = 1. The...

Let the normal at a point P on the curve y2 – 3x2 + y + 10 = 0 intersect the y-axis at $$\left( {0,{3 \over 2}} \right)$$ . If m is the slope of the t...

If the function $$f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a}> 0$$ has a local maximum at $$x=\alpha$$ and a local minimum at $$x=\alph...

Let $$f(x)=4 \cos ^3 x+3 \sqrt{3} \cos ^2 x-10$$. The number of points of local maxima of $$f$$ in interval $$(0,2 \pi)$$ is

The number of critical points of the function $$f(x)=(x-2)^{2 / 3}(2 x+1)$$ is

For the function $$f(x)=(\cos x)-x+1, x \in \mathbb{R}$$, between the following two statements (S1) $$f(x)=0$$ for only one value of $$x$$ in $$[0, \p...

The interval in which the function $$f(x)=x^x, x>0$$, is strictly increasing is

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rect...

Let $$f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}$$, and $$g(x)$$ be a function such that $$g(f(x))=x$$ for all $$x \in \mathbf{R}$$. Then $$\frac{g(7)}{g^...

For the function $$f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],$$ consider the following two s...

Let $$f(x)=3 \sqrt{x-2}+\sqrt{4-x}$$ be a real valued function. If $$\alpha$$ and $$\beta$$ are respectively the minimum and the maximum values of $$f...

Let the sum of the maximum and the minimum values of the function $$f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}$$ be $$\frac{m}{n}$$, where $$\operatorname{g...

If $5 f(x)+4 f\left(\frac{1}{x}\right)=x^2-2, \forall x \neq 0$ and $y=9 x^2 f(x)$, then $y$ is strictly increasing in :

Let $$f: \rightarrow \mathbb{R} \rightarrow(0, \infty)$$ be strictly increasing function such that $$\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}...

If the function $$f:(-\infty,-1] \rightarrow(a, b]$$ defined by $$f(x)=e^{x^3-3 x+1}$$ is one - one and onto, then the distance of the point $$P(2 b+4...

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } ...

Let $$f(x)=(x+3)^2(x-2)^3, x \in[-4,4]$$. If $$M$$ and $$m$$ are the maximum and minimum values of $$f$$, respectively in $$[-4,4]$$, then the value o...

The maximum area of a triangle whose one vertex is at $$(0,0)$$ and the other two vertices lie on the curve $$y=-2 x^2+54$$ at points $$(x, y)$$ and $...

The function $$f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}$$

The function $$f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}$$, has

Consider the function $$f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}$$ defined by $$f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1$$. Consider the stateme...

Let $$g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)$$ and $$f^{\prime \prime}(x)>0$$ for all $$x \in(0,3)$$. If $$g$$ is decreasing in $$(0, \alpha)$$ and i...

$$\max _\limits{0 \leq x \leq \pi}\left\{x-2 \sin x \cos x+\frac{1}{3} \sin 3 x\right\}=$$

If the local maximum value of the function $$f(x)=\left(\frac{\sqrt{3 e}}{2 \sin x}\right)^{\sin ^{2} x}, x \in\left(0, \frac{\pi}{2}\right)$$ , is $$...

Let $$f:[2,4] \rightarrow \mathbb{R}$$ be a differentiable function such that $$\left(x \log _{e} x\right) f^{\prime}(x)+\left(\log _{e} x\right) f(x)...

Let $$\mathrm{g}(x)=f(x)+f(1-x)$$ and $$f^{\prime \prime}(x) > 0, x \in(0,1)$$. If $$\mathrm{g}$$ is decreasing in the interval $$(0, a)$$ and increas...

The slope of tangent at any point (x, y) on a curve $$y=y(x)$$ is $${{{x^2} + {y^2}} \over {2xy}},x > 0$$. If $$y(2) = 0$$, then a value of $$y(8)$$ i...

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. ...

The sum of the absolute maximum and minimum values of the function $$f(x)=\left|x^{2}-5 x+6\right|-3 x+2$$ in the interval $$[-1,3]$$ is equal to :...

A wire of length $$20 \mathrm{~m}$$ is to be cut into two pieces. A piece of length $$l_{1}$$ is bent to make a square of area $$A_{1}$$ and the other...

If the functions $f(x)=\frac{x^3}{3}+2 b x+\frac{a x^2}{2}$ and $g(x)=\frac{x^3}{3}+a x+b x^2, a \neq 2 b$ have a common extreme point, then $a+2 b+7...

The number of points on the curve $$y=54 x^{5}-135 x^{4}-70 x^{3}+180 x^{2}+210 x$$ at which the normal lines are parallel to $$x+90 y+2=0$$ is :...

Let the function $$f(x) = 2{x^3} + (2p - 7){x^2} + 3(2p - 9)x - 6$$ have a maxima for some value of $$x 0$$. Then, the set of all values of p is...

Let $$x=2$$ be a local minima of the function $$f(x)=2x^4-18x^2+8x+12,x\in(-4,4)$$. If M is local maximum value of the function $$f$$ in ($$-4,4)$$, t...

Let $$f:(0,1)\to\mathbb{R}$$ be a function defined $$f(x) = {1 \over {1 - {e^{ - x}}}}$$, and $$g(x) = \left( {f( - x) - f(x)} \right)$$. Consider two...

Let $$f(x)=3^{\left(x^{2}-2\right)^{3}+4}, x \in \mathrm{R}$$. Then which of the following statements are true? $$\mathrm{P}: x=0$$ is a point of loca...

The function $$f(x)=x \mathrm{e}^{x(1-x)}, x \in \mathbb{R}$$, is :

If the minimum value of $$f(x)=\frac{5 x^{2}}{2}+\frac{\alpha}{x^{5}}, x>0$$, is 14 , then the value of $$\alpha$$ is equal to :

If the maximum value of $$a$$, for which the function $$f_{a}(x)=\tan ^{-1} 2 x-3 a x+7$$ is non-decreasing in $$\left(-\frac{\pi}{6}, \frac{\pi}{6}\r...

If the absolute maximum value of the function $$f(x)=\left(x^{2}-2 x+7\right) \mathrm{e}^{\left(4 x^{3}-12 x^{2}-180 x+31\right)}$$ in the interval $$...

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

If xy4 attains maximum value at the point (x, y) on the line passing through the points (50 + $$\alpha$$, 0) and (0, 50 + $$\alpha$$), $$\alpha$$ > 0,...

Let $$f(x) = 4{x^3} - 11{x^2} + 8x - 5,\,x \in R$$. Then f :

Let f : R $$\to$$ R be a function defined by f(x) = (x $$-$$ 3)n1 (x $$-$$ 5)n2, n1, n2 $$\in$$ N. Then, which of the following is NOT true?...

A wire of length 22 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into an equilateral triangle. Then, th...

The number of real solutions of $${x^7} + 5{x^3} + 3x + 1 = 0$$ is equal to ____________.

Consider a cuboid of sides 2x, 4x and 5x and a closed hemisphere of radius r. If the sum of their surface areas is a constant k, then the ratio x : r,...

The sum of the absolute minimum and the absolute maximum values of the function f(x) = |3x $$-$$ x2 + 2| $$-$$ x in the interval [$$-$$1, 2] is :...

Let S be the set of all the natural numbers, for which the line $${x \over a} + {y \over b} = 2$$ is a tangent to the curve $${\left( {{x \over a}} \r...

Let $$f(x) = 2{\cos ^{ - 1}}x + 4{\cot ^{ - 1}}x - 3{x^2} - 2x + 10$$, $$x \in [ - 1,1]$$. If [a, b] is the range of the function f, then 4a $$-$$ b i...

Water is being filled at the rate of 1 cm3 / sec in a right circular conical vessel (vertex downwards) of height 35 cm and diameter 14 cm. When the he...

If the angle made by the tangent at the point (x0, y0) on the curve $$x = 12(t + \sin t\cos t)$$, $$y = 12{(1 + \sin t)^2}$$, $$0 0 is equal to:...

The slope of normal at any point (x, y), x > 0, y > 0 on the curve y = y(x) is given by $${{{x^2}} \over {xy - {x^2}{y^2} - 1}}$$. If the curve passes...

Let $$\lambda$$$$^ * $$ be the largest value of $$\lambda$$ for which the function $${f_\lambda }(x) = 4\lambda {x^3} - 36\lambda {x^2} + 36x + 48$$ i...

The surface area of a balloon of spherical shape being inflated, increases at a constant rate. If initially, the radius of balloon is 3 units and afte...

For the function $$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$$, which one of the following is NOT correct?

If the tangent at the point (x1, y1) on the curve $$y = {x^3} + 3{x^2} + 5$$ passes through the origin, then (x1, y1) does NOT lie on the curve :...

The sum of absolute maximum and absolute minimum values of the function $$f(x) = |2{x^2} + 3x - 2| + \sin x\cos x$$ in the interval [0, 1] is :

Let $$\lambda x - 2y = \mu $$ be a tangent to the hyperbola $${a^2}{x^2} - {y^2} = {b^2}$$. Then $${\left( {{\lambda \over a}} \right)^2} - {\left( {...

The function $$f(x) = {x^3} - 6{x^2} + ax + b$$ is such that $$f(2) = f(4) = 0$$. Consider two statements :Statement 1 : there exists x1, x2 $$\in$$(2...

The number of real roots of the equation $${e^{4x}} + 2{e^{3x}} - {e^x} - 6 = 0$$ is :

A box open from top is made from a rectangular sheet of dimension a $$\times$$ b by cutting squares each of side x from each of the four corners and f...

A wire of length 20 m is to be cut into two pieces. One of the pieces is to be made into a square and the other into a regular hexagon. Then the lengt...

The local maximum value of the function $$f(x) = {\left( {{2 \over x}} \right)^{{x^2}}}$$, x > 0, is

Let $$f(x) = 3{\sin ^4}x + 10{\sin ^3}x + 6{\sin ^2}x - 3$$, $$x \in \left[ { - {\pi \over 6},{\pi \over 2}} \right]$$. Then, f is :

Let f : R $$\to$$ R be defined as$$f(x) = \left\{ {\matrix{ { - {4 \over 3}{x^3} + 2{x^2} + 3x,} & {x > 0} \cr {3x{e^x},} & {x \le ...

The sum of all the local minimum values of the twice differentiable function f : R $$\to$$ R defined by $$f(x) = {x^3} - 3{x^2} - {{3f''(2)} \over 2}x...

Let $$A = [{a_{ij}}]$$ be a 3 $$\times$$ 3 matrix, where $${a_{ij}} = \left\{ {\matrix{ 1 & , & {if\,i = j} \cr { - x} & , & {...

Let 'a' be a real number such that the function f(x) = ax2 + 6x $$-$$ 15, x $$\in$$ R is increasing in $$\left( { - \infty ,{3 \over 4}} \right)$$ and...

Consider the function f : R $$ \to $$ R defined by $$f(x) = \left\{ \matrix{ \left( {2 - \sin \left( {{1 \over x}} \right)} \right)|x|,x \ne 0 \hfi...

Let f be a real valued function, defined on R $$-$$ {$$-$$1, 1} and given by f(x) = 3 loge $$\left| {{{x - 1} \over {x + 1}}} \right| - {2 \over {x - ...

The maximum value of $$f(x) = \left| {\matrix{ {{{\sin }^2}x} & {1 + {{\cos }^2}x} & {\cos 2x} \cr {1 + {{\sin }^2}x} & {{{\cos }^...

Let slope of the tangent line to a curve at any point P(x, y) be given by $${{x{y^2} + y} \over x}$$. If the curve intersects the line x + 2y = 4 at x...

The maximum slope of the curve $$y = {1 \over 2}{x^4} - 5{x^3} + 18{x^2} - 19x$$ occurs at the point :

Let f be any function defined on R and let it satisfy the condition : $$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$$If f(0) = 1, then :...

If the curves, $${{{x^2}} \over a} + {{{y^2}} \over b} = 1$$ and $${{{x^2}} \over c} + {{{y^2}} \over d} = 1$$ intersect each other at an angle of 90$...

If Rolle's theorem holds for the function $$f(x) = {x^3} - a{x^2} + bx - 4$$, $$x \in [1,2]$$ with $$f'\left( {{4 \over 3}} \right) = 0$$, then ordere...

For which of the following curves, the line $$x + \sqrt 3 y = 2\sqrt 3 $$ is the tangent at the point $$\left( {{{3\sqrt 3 } \over 2},{1 \over 2}} \ri...

Let $$f:R \to R$$ be defined as$$f(x) = \left\{ {\matrix{ { - 55x,} & {if\,x < - 5} \cr {2{x^3} - 3{x^2} - 120x,} & {if\, - 5 \le ...

If the curve y = ax2 + bx + c, x$$ \in $$R, passes through the point (1, 2) and the tangent line to this curve at origin is y = x, then the possible v...

The function f(x) = $${{4{x^3} - 3{x^2}} \over 6} - 2\sin x + \left( {2x - 1} \right)\cos x$$ :

If the tangent to the curve y = x3 at the point P(t, t3) meets the curve again at Q, then the ordinate of the point which divides PQ internally in the...

The set of all real values of $$\lambda $$ for which the function $$f(x) = \left( {1 - {{\cos }^2}x} \right)\left( {\lambda + \sin x} \right),x \in \...

If the tangent to the curve, y = f (x) = xloge x, (x > 0) at a point (c, f(c)) is parallel to the line-segment joining the points (1, 0) and (e, e)...

The position of a moving car at time t is given by f(t) = at2 + bt + c, t > 0, where a, b and c are real numbers greater than 1. Then the average s...

Which of the following points lies on the tangent to the curve x4ey + 2$$\sqrt {y + 1} $$ = 3 at the point (1, 0)?

If x = 1 is a critical point of the function f(x) = (3x2 + ax – 2 – a)ex , then :

If the point P on the curve, 4x2 + 5y2 = 20 is farthest from the point Q(0, -4), then PQ2 is equal to:

The area (in sq. units) of the largest rectangle ABCD whose vertices A and B lie on the x-axis and vertices C and D lie on the parabola, y = x2–1 belo...

Let f be a twice differentiable function on (1, 6). If f(2) = 8, f’(2) = 5, f’(x) $$ \ge $$ 1 and f''(x) $$ \ge $$ 4, for all x $$ \in $$ (1, 6), then...

If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape; then the rate of change of its volume (in cm3/sec), when th...

The function, f(x) = (3x – 7)x2/3, x $$ \in $$ R, is increasing for all x lying in :

Let f : (–1, $$\infty $$) $$ \to $$ R be defined by f(0) = 1 and f(x) = $${1 \over x}{\log _e}\left( {1 + x} \right)$$, x $$ \ne $$ 0. Then the functi...

The equation of the normal to the curve y = (1+x)2y + cos 2(sin–1x) at x = 0 is :

If p(x) be a polynomial of degree three that has a local maximum value 8 at x = 1 and a local minimum value 4 at x = 2; then p(0) is equal to :

Let P(h, k) be a point on the curve y = x2 + 7x + 2, nearest to the line, y = 3x – 3. Then the equation of the normal to the curve at P is :...

If the tangent to the curve y = x + sin y at a point (a, b) is parallel to the line joining $$\left( {0,{3 \over 2}} \right)$$ and $$\left( {{1 \over ...

A spherical iron ball of 10 cm radius is coated with a layer of ice of uniform thickness the melts at a rate of 50 cm3/min. When the thickness of ice ...

The length of the perpendicular from the origin, on the normal to the curve, x2 + 2xy – 3y2 = 0 at the point (2,2) is

If c is a point at which Rolle's theorem holds for the function, f(x) = $${\log _e}\left( {{{{x^2} + \alpha } \over {7x}}} \right)$$ in the interval [...

Let ƒ(x) = xcos–1(–sin|x|), $$x \in \left[ { - {\pi \over 2},{\pi \over 2}} \right]$$, then which of the following is true?

The value of c in the Lagrange's mean value theorem for the function ƒ(x) = x3 - 4x2 + 8x + 11, when x $$ \in $$ [0, 1] is: ...

Let ƒ(x) be a polynomial of degree 5 such that x = ±1 are its critical points. If $$\mathop {\lim }\limits_{x \to 0} \left( {2 + {{f\left( x \right)} ...

Let the function, ƒ:[-7, 0]$$ \to $$R be continuous on [-7,0] and differentiable on (-7, 0). If ƒ(-7) = - 3 and ƒ'(x) $$ \le $$ 2, for all x $$ \in $$...

If m is the minimum value of k for which the function f(x) = x$$\sqrt {kx - {x^2}} $$ is increasing in the interval [0,3] and M is the maximum value o...

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/sec, then the rate (in cm/sec.) a...

If the tangent to the curve $$y = {x \over {{x^2} - 3}}$$ , $$x \in \rho ,\left( {x \ne \pm \sqrt 3 } \right)$$, at a point ($$\alpha $$, $$\beta $$)...

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm3 /min. When the thickness of th...

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $${\tan ^{ - 1}}\left( {{1 \over 2}} \right)$$. Water is p...

If ƒ(x) is a non-zero polynomial of degree four, having local extreme points at x = –1, 0, 1; then the set S = {x $$ \in $$ R : ƒ(x) = ƒ(0)} Contains...

Let S be the set of all values of x for which the tangent to the curve y = ƒ(x) = x3 – x2 – 2x at (x, y) is parallel to the line segment joining the ...

If the tangent to the curve, y = x3 + ax – b at the point (1, –5) is perpendicular to the line, –x + y + 4 = 0, then which one of the following points...

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is

Given that the slope of the tangent to a curve y = y(x) at any point (x, y) is $$2y \over x^2$$. If the curve passes through the centre of the circle...

Let ƒ : [0, 2] $$ \to $$ R be a twice differentiable function such that ƒ''(x) > 0, for all x $$ \in $$ (0, 2). If $$\phi $$(x) = ƒ(x) + ƒ(2 – x), ...

If S1 and S2 are respectively the sets of local minimum and local maximum points of the function, ƒ(x) = 9x4 + 12x3 – 36x2 + 25, x $$ \in $$ R, then ...

If the function f given by f(x) = x3 – 3(a – 2)x2 + 3ax + 7, for some a$$ \in $$R is increasing in (0, 1] and decreasing in [1, 5), then a root of th...

The tangent to the curve y = x2 – 5x + 5, parallel to the line 2y = 4x + 1, also passes through the point :

Let f(x) = $${x \over {\sqrt {{a^2} + {x^2}} }} - {{d - x} \over {\sqrt {{b^2} + {{\left( {d - x} \right)}^2}} }},\,\,$$ x $$\, \in $$ R, where a, b a...

The maximum value of the function f(x) = 3x3 – 18x2 + 27x – 40 on the set S = $$\left\{ {x\, \in R:{x^2} + 30 \le 11x} \right\}$$ is :...

The tangent to the curve, y = xex2 passing through the point (1, e) also passes through the point

A helicopter is flying along the curve given by y – x3/2 = 7, (x $$ \ge $$ 0). A soldier positioned at the point $$\left( {{1 \over 2},7} \right)$$ wa...

The shortest distance between the point  $$\left( {{3 \over 2},0} \right)$$   and the curve y = $$\sqrt x $$, (x > 0), is -

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is :

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, f(x) = 2x3 $$-$$ 9x2 + 12x + 5 in the interval [0, ...

Let $$f\left( x \right) = {x^2} + {1 \over {{x^2}}}$$ and $$g\left( x \right) = x - {1 \over x}$$, $$x \in R - \left\{ { - 1,0,1} \right\}$$. If $$h\...

If the curves y2 = 6x, 9x2 + by2 = 16 intersect each other at right angles, then the value of b is :

If $$\beta $$ is one of the angles between the normals to the ellipse, x2 + 3y2 = 9 at the points (3 cos $$\theta $$, $$\sqrt 3 \sin \theta $$) and (...

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm2) of this cone is : ...

The function f defined by f(x) = x3 $$-$$ 3x2 + 5x + 7 , is :

A tangent to the curve, y = f(x) at P(x, y) meets x-axis at A and y-axis at B. If AP : BP = 1 : 3 and f(1) = 1, then the curve also passes through the...

The tangent at the point (2, $$-$$2) to the curve, x2y2 $$-$$ 2x = 4(1 $$-$$ y) does not pass through the point :

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 :

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 flower-bed, ...

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

Let f(x) = sin4x + cos4 x. Then f is an increasing function in the interval :

The minimum distance of a point on the curve y = x2−4 from the origin is :

If the tangent at a point P, with parameter t, on the curve x = 4t2 + 3, y = 8t3−1, t $$ \in $$ R, meets the curve again at a point Q, then the coordi...

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 $$=r$$ uni...

Consider : f $$\left( x \right) = {\tan ^{ - 1}}\left( {\sqrt {{{1 + \sin x} \over {1 - \sin x}}} } \right),x \in \left( {0,{\pi \over 2}} \right).$$...

Let $$f(x)$$ be a polynomial of degree four having extreme values at $$x=1$$ and $$x=2$$. If $$\mathop {\lim }\limits_{x \to 0} \left[ {1 + {{f\left(...

The normal to the curve, $${x^2} + 2xy - 3{y^2} = 0$$, at $$(1,1)$$

If $$f$$ and $$g$$ are differentiable functions in $$\left[ {0,1} \right]$$ satisfying $$f\left( 0 \right) = 2 = g\left( 1 \right),g\left( 0 \right) ...

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

The real number $$k$$ for which the equation, $$2{x^3} + 3x + k = 0$$ has two distinct real roots in $$\left[ {0,\,1} \right]$$

The intercepts on $$x$$-axis made by tangents to the curve, $$y = \int\limits_0^x {\left| t \right|dt,x \in R,} $$ which are parallel to the line $$y...

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 $...

Let $$a,b \in R$$ be such that the function $$f$$ given by $$f\left( x \right) = In\left| x \right| + b{x^2} + ax,\,x \ne 0$$ has extreme values at $$...

A line is drawn through the point $$(1, 2)$$ to meet the coordinate axes at $$P$$ and $$Q$$ such that it forms a triangle $$OPQ,$$ where $$O$$ is the...

For $$x \in \left( {0,{{5\pi } \over 2}} \right),$$ define $$f\left( x \right) = \int\limits_0^x {\sqrt t \sin t\,dt.} $$ Then $$f$$ has

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

Let $$f:R \to R$$ be defined by $$$f\left( x \right) = \left\{ {\matrix{ {k - 2x,\,\,if} & {x \le - 1} \cr {2x + 3,\,\,if} & {x >...

Let $$f:R \to R$$ be a continuous function defined by $$$f\left( x \right) = {1 \over {{e^x} + 2{e^{ - x}}}}$$$ Statement - 1 : $$f\left( c \right) =...

The equation of the tangent to the curve $$y = x + {4 \over {{x^2}}}$$, that is parallel to the $$x$$-axis, is

Given $$P\left( x \right) = {x^4} + a{x^3} + b{x^2} + cx + d$$ such that $$x=0$$ is the only real root of $$P'\,\left( x \right) = 0.$$ If $$P\left( ...

How many real solutions does the equation $${x^7} + 14{x^5} + 16{x^3} + 30x - 560 = 0$$ have?

Suppose the cubic $${x^3} - px + q$$ has three distinct real roots where $$p>0$$ and $$q>0$$. Then which one of the following holds?

The function $$f\left( x \right) = {\tan ^{ - 1}}\left( {\sin x + \cos x} \right)$$ is an incresing function in

A value of $$c$$ for which conclusion of Mean Value Theorem holds for the function $$f\left( x \right) = {\log _e}x$$ on the interval $$\left[ {1,3} \...

If $$p$$ and $$q$$ are positive real numbers such that $${p^2} + {q^2} = 1$$, then the maximum value of $$(p+q)$$ is

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

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $$...

Angle between the tangents to the curve $$y = {x^2} - 5x + 6$$ at the points $$(2,0)$$ and $$(3,0)$$ is

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?

If the equation $${a_n}{x^n} + {a_{n - 1}}{x^{n - 1}} + ........... + {a_1}x = 0$$ $${a_1} \ne 0,n \ge 2,$$ has a positive root $$x = \alpha $$, then ...

Let f be differentiable for all x. If f(1) = -2 and f'(x) $$ \ge $$ 2 for x $$ \in \left[ {1,6} \right]$$, then

Area of the greatest rectangle that can be inscribed in the ellipse $${{{x^2}} \over {{a^2}}} + {{{y^2}} \over {{b^2}}} = 1$$

The normal to the curve $$x = a\left( {\cos \theta + \theta \sin \theta } \right),y = a\left( {\sin \theta - \theta \cos \theta } \right)$$ at any ...

A spherical iron ball $$10$$ cm in radius is coated with a layer of ice of uniform thickness that melts at a rate of $$50$$ cm$$^3$$ /min. When the th...

A lizard, at an initial distance of 21 cm behind an insect moves from rest with an acceleration of $2 \mathrm{~cm} / \mathrm{s}^2$ and pursues the ins...

A point on the parabola $${y^2} = 18x$$ at which the ordinate increases at twice the rate of the abscissa is

A function $$y=f(x)$$ has a second order derivative $$f''\left( x \right) = 6\left( {x - 1} \right).$$ If its graph passes through the point $$(2, 1)$...

If $$2a+3b+6c=0$$, then at least one root of the equation $$a{x^2} + bx + c = 0$$ lies in the interval

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

The real number $$x$$ when added to its inverse gives the minimum sum at $$x$$ equal :

If the function $$f\left( x \right) = 2{x^3} - 9a{x^2} + 12{a^2}x + 1,$$ where $$a>0,$$ attains its maximum and minimum at $$p$$ and $$q$$ respecti...

The maximum distance from origin of a point on the curve $$x = a\sin t - b\sin \left( {{{at} \over b}} \right)$$ $$y = a\cos t - b\cos \left( {{{at} ...

If $$2a+3b+6c=0,$$ $$\left( {a,b,c \in R} \right)$$ then the quadratic equation $$a{x^2} + bx + c = 0$$ has