Application of Derivatives — JEE Main PYQs

Let $f:[-1,1]\to R$ be defined as $f(x)=a{x}^2+bx+c$ for all $x\in [-1,1]$, where $a,b,c\in R$ such that $f(-1)=2$, $f^{\prime }(-1)=1$ and for $x\in (-1,1)$ the maximum value of $f^{\prime \prime }(x)$ is $\frac{1}{2}$. If $f(x)\le \alpha$, $x\in [-1,1]$, then the …

Let the function, $f:[-7,0]\to \mathbb{R}$ be continuous on $[-7,0]$ and differentiable on $(-7,0)$. If $f(-7)=-3$ and $f^{\prime }(x)\le 2$, for all $x\in (-7,0)$, then for all such functions $f$, $f(-1)+f(0)$ lies in the interval:

The value of $c$ in the Lagrange's mean value theorem for the function $f(x)=x^3-4x^2+8x+11$, when $x\in [0,1]$ is:

Let $f(x)$ be a polynomial of degree 5 such that $x=\pm 1$ are its critical points. If ${\lim }_{x\to 0}\left(2+\frac{f(x)}{x^3}\right)=4$, then which one of the following is not true?

If $c$ is a point at which Rolle's theorem holds for the function, $f(x)={\log }_e\left(\frac{x^2+\alpha }{7x}\right)$ in the interval $[3,4]$, where $\alpha \in R$, then $f^{\prime \prime }(c)$ is equal to:

Let the normal at a point $P$ on the curve $y^2-3x^2+y+10=0$ intersect the $y$ - axis at $\left(0,\frac{3}{2}\right)$. If m is the slope of the tangent at P to the curve, then $|\mathrm{m}|$ is equal to

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

Let $f(x)$ be a polynomial of degree 3 such that $f(-1)=10,f(1)=-6,f(x)$ has a critical point at $x=-1$ and $f^{\prime }(x)$ has a critical point at $x=1$. Then $f(x)$ has a local minima at $x=$

Let a function $f:[0,5]\to R$ be continuous, $f(1)=3$ and $F$ be defined as:
$F(x)={\int }_1^x{t}^{2}g(t)dt$, where $g(t)={\int }_1^{t}f(u)du$. Then for the function $F$, the point $x=1$ is:

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

Let $f$ be any function continuous on $[a,b]$ and twice differentiable on (a, b). If for all $x\in (a,b),f^{\prime }(x)>0$ and $f^{\prime \prime }(x)<0$, then for any $c\in (a,b)$ $\frac{f(c)-f(a)}{f(b)-f(c)}$ is greater than:

If $x=1$ is a critical point of the function $f(x)=\left(3x^2+ax-2-a\right)e^x$, then

If the lines $x+y=a$ and $x-y=b$ touch the curve $y=x^2-3x+2$ at the points where the curve intersects the $x$-axis, then $\frac{a}{b}$ is equal to ____.

The function, $f(x)=(3x-7)x^{2/3},x\in R$, is increasing for all $x$ lying in:

If the surface area of a cube is increasing at a rate of $3.6{\mathrm{cm}}^2/\sec$, retaining its shape; then the rate of change of its volume (in ${\mathrm{cm}}^3/\sec$ ), when the length of a side of the cube is 10 cm , is:

Suppose $f(x)$ is a polynomial of degree four, having critical points at $-1,0,1$. If $T=\{x\in R\mid f(x)=f(0)\}$, then the sum of squares of all the elements of $T$ is:

If the tangent to the curve, $\mathrm{y}={\mathrm{e}}^{\mathrm{x}}$ at a point ( $\mathrm{c},{\mathrm{e}}^{\mathrm{c}}$ ) and the normal to the parabola, $y^2=4x$ at the point $(1,2)$ intersect at the same point on the $x$-axis, then the value of $c$ is ______.

The set of all real values of $\lambda$ for which the function $f(x)=\left(1-{\cos }^{2}x\right)\cdot (\lambda +\sin x)$, $\mathrm{x}\in \left(-\frac{\pi }{2},\frac{\pi }{2}\right)$, has exactly one maxima and exactly one minima, is

For all twice differentiable functions
$f:R\to R$, with $f(0)=f(1)=f^{\prime }(0)=0$,

If the tangent to the curve, $y=f(x)=x{\log }_{e}x,(x>0)$ at a point ( $c,f(c)$ ) is parallel to the line - segment joining the points $(1,0)$ and $(e,e)$ then $c$ is equal to:

The position of a moving car at time $t$ is given by $f(t)=a{t}^2+bt+c,t>0$, where $a,b$ and c are real numbers greater than 1. Then the average speed of the car over the time interval $\left[t_1,t_2\right]$ is attained at the point:

Let $AD$ and $BC$ be two vertical poles at $A$ and $b$ respectively on a horizontal ground. If $AD=8\mathrm{m},BC=11\mathrm{m}$ and $AB=10\mathrm{m}$; then the distance (in meters) of a point $M$ on $AB$ from the point $A$ such that …

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

Let $x_0$ be the point of local maxima of $f(x)=\vec{a}\cdot (\vec{b}\times \vec{c})$ where $\vec{a}=x\hat{i}-2\hat{j}+3\hat{k},\vec{b}=-2\hat{i}+x\hat{j}-\hat{k}$ and $\vec{c}=7\hat{i}-2\hat{j}+x\hat{k}$. Then the value of …

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, $y=x^2-1$ below the $x$-axis, is:

The minimum value of $2^{\sin x}+2^{\cos x}$ is:

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

If ${\mathrm{S}}_1$ and ${\mathrm{S}}_2$ are respectively the sets of local minimum and local maximum points of the function. $f(x)=9x^4+12x^3-36x^2+25,x\in R$, then

Let $\mathrm{f}:[0,2]\to \mathrm{R}$ be a twice differentiable function such that ${\mathrm{f}}^{\prime \prime }(\mathrm{x})>0$, for all $\mathrm{x}\in (0,2)$. If $\varphi (x)=f(x)+f(2-x)$, then $\varphi$ is:

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

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

Let $S$ be the set of all values of $x$ for which the tangent to the curve $y=f(x)=x^3-x^2-2x$ at ( $x,y$ ) is parallel to the line segment joining the points ( $1,f(1)$ ) and ( $-1,f(-1)$ ), then $S$ is equal to:

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

Let $x$, $y$ be positive real numbers and $m$, $n$ positive integers. The maximum value of the expression $\frac{x^m{y}^n}{(1+x^{2m})(1+y^{2n})}$ is:

Let $f(x)=\frac{x}{\sqrt{a^2+x^2}}-\frac{d-x}{\sqrt{b^2+(d-x{)}^2}}$, $x\in R$, where $a,b$ and $d$ are non-zero real constants. Then:

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 of $f$ in $[0,3]$ when $k=m$, then the ordered pair $(\mathrm{m},\mathrm{M})$ is equal to:

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate $25\mathrm{cm}/\sec$., then the rate (in $\mathrm{cm}/\sec$.) at which the bottom of the ladder slides away from the wall on the horizontal gro …

If the tangent to the curve $y=\frac{x}{x^2-3},x\in R,(x\ne \pm \sqrt{3})$ at a point $(\alpha ,\beta )\ne (0,0)$ on it is parallel to the line $2x+6y-11=0$ then

Let $a_1,a_2,a_3,\dots \dots$ be and A.P with $a_6=2$. Then the common difference of this A.P., which maximizes the product ${\mathrm{a}}_1{\mathrm{a}}_4{\mathrm{a}}_5$ is

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

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

The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, $y=12-x^2$ such that the rectangle lies inside the parabola, is:

The tangents to the curve $\mathrm{y}=(\mathrm{x}-2{)}^2-1$ at its points of intersection with the line $\mathrm{x}-\mathrm{y}=3$, intersect at the point :

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

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

A water tank has the shape of an inverted right circular cone, whose semi vertical angle is ${\tan }^{-1}\left(\frac{1}{2}\right)$. Water is poured in at a constant rate of 5 cubic meter per minute. Then the rate (in $\mathrm{m}/\min$) at which the level of water is rising …

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

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

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

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