Application of Derivatives — JEE Main PYQs

Let $f(x)=x^2+\frac{1}{x^2}$ and $\mathrm{g}(x)=x-\frac{1}{x}$, $x\in \mathbf{R}-\{-1,0,1\}$. If $\mathrm{h}(x)=\frac{f(x)}{\mathrm{g}(x)}$, then the local minimum value of $\mathrm{h}(x)$ is :

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, is :

The radius of a circle, having minimum area, which touches the curve $y=4-x^2$ and the lines, $y=|x|$ 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 $=r$ units. If the sum of the areas of the square and the circle so formed is minimum, then:

Consider $f(x)={\tan }^{-1}\left(\sqrt{\frac{1+\sin x}{1-\sin x}}\right),x\in \left(0,\frac{\pi }{2}\right)$. A normal to $y=f(x)$ at $x=\frac{\pi }{6}$ also passes through the point:

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree four having extreme values at $\mathrm{x}=1$ and $\mathrm{x}=2$. If ${\lim }_{\mathrm{x}\to 0}\left[1+\frac{\mathrm{f}(\mathrm{x})}{{\mathrm{x}}^2}\right]=3$, then $f(2)$ is equal to :

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

If $f$ and $g$ are differentiable functions in $[0,1]$ satisfying $\mathrm{f}(0)=2=g(1),g(0)=0$ and $f(1)=6$, then for some $\mathrm{c}\in ]0,1[$

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

The intercepts on x -axis made by tangents to the curve, $\mathrm{y}={\int }_0^{\mathrm{x}}|\mathrm{t}|\mathrm{dt},\mathrm{x}\in \mathrm{R}$, which are parallel to the line $\mathrm{y}=2\mathrm{x}$, are equal to

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

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 …

Let $a,b\in R$ be such that the function $f$ given by $f(x)=\ln |x|+b{x}^2+ax,x\ne 0$ has extreme values at $x=-1$ and $x=2$.

Statement 1: f has local maximum at $\mathrm{x}=-1$ and at $\mathrm{x}=2$.
Statement 2: $\mathrm{a}=\frac{1}{2}$ and …

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 origin. If the area of the triangle OPQ is least, then the slope of the line PQ is

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

For $x\in \left(0,\frac{5\pi }{2}\right)$, define $f(x)={\int }_0^x\sqrt{t}\sin tdt$. Then $f$ has

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

Let $f:R\to R$ be defined by $f(x)=\{\begin{array}{ll}k-2x,& \text{ if }x\le -1\\ 2x+3,& \text{ if }x>-1\end{array}$. If $f$ has a local minimum at $x=-1$, then a possible value of k is

Given $\mathrm{P}(\mathrm{x})={\mathrm{x}}^4+{\mathrm{ax}}^3+{\mathrm{bx}}^2+\mathrm{cx}+\mathrm{d}$ such that $\mathrm{x}=0$ is the only real root of ${\mathrm{P}}^{\prime }(\mathrm{x})=0$. If $\mathrm{P}(-1)<\mathrm{P}(1)$, then in the interval $[-1,1]$

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

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?

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

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

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f(x)={\log }_{e}x$ on the interval $[1,3]$ is

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

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

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

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 x. The maximum area enclosed by the park is

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

The normal to the curve $x=a(\cos \theta +\theta \sin \theta ),y=a(\sin \theta -\theta \cos \theta )$ at any point '$\theta$' is such that

A function is matched below against an interval where it is supposed to be increasing. Which of the following pairs is incorrectly matched?
| (2) $[2,\infty )$ | …

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

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

Two points $A$ and $B$ move from rest along a straight line with constant acceleration $f$ and $f^{\prime }$ respectively. If $A$ takes $m$ sec. more than $B$ and describes '$n$' units more than $B$ in acquiring the same speed then

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

If $u=\sqrt{a^2{\cos }^2\theta +b^2{\sin }^2\theta }+\sqrt{a^2{\sin }^2\theta +b^2{\cos }^2\theta }$, then the difference between the maximum and minimum values of $u^2$ is given by

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 $f^{\prime \prime }(x)=6(x-1)$. If its graph passes through the point ( 2,1 ) and at that point the tangent to the graph is $y=3x-5$, then the function is

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