Application of Derivatives — JEE Main PYQs

Let $f:R\to R$ be a twice differentiable function such that the quadratic equation $\mathrm{f}(\mathrm{x}){\mathrm{m}}^2-2{\mathrm{f}}^{\prime }(\mathrm{x})\mathrm{m}+{\mathrm{f}}^{\prime }(\mathrm{x})=0$ in m , has two equal roots for every …

Let $f:\mathbf{R}\to \mathbf{R}$ be a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbf{R}$ and $f^{\prime }(a-1)=0$, where a is real number. Let $g(x)=f\left({\tan }^{2}x-2\tan x+a\right)$, $0<x<\frac{\pi }{2}$.

Consider the following two statements:
(I) g i …

Let the maximum value of ${\left({\sin }^{-1}\mathrm{x}\right)}^2+{\left({\cos }^{-1}\mathrm{x}\right)}^2$ for $x\in \left[-\frac{\sqrt{3}}{2},\frac{1}{\sqrt{2}}\right]$ be $\frac{m}{n}{\pi }^2$, where $\gcd (m,n)=1$.

Then $\mathrm{m}+\mathrm{n}$ is equal to ____.

Let $f(x)=x^{2025}-x^{2000},x\in [0,1]$ and the minimum value of the function $\mathrm{f}(\mathrm{x})$ in the interval $[0,1]$ be $(80{)}^{80}(\mathrm{n}{)}^{-81}$. Then n is equal to

Let $(2\alpha ,\alpha )$ be the largest interval in which the function $f(t)=\frac{|t+1|}{t^2},t<0$, is strictly decreasing. Then the local maximum value of the function
$g(x)=2{\log }_e(x-2)+\alpha x^2+4x-\alpha ,x>2$, is ______

Let f be a differentiable function satisfying
$\mathrm{f}(\mathrm{x})=1-2\mathrm{x}+{\int }_0^{\mathrm{x}}{\mathrm{e}}^{(\mathrm{x}-\mathrm{t})}\mathrm{f}(\mathrm{t})\mathrm{dt},\mathrm{x}\in \mathbf{R}$ and let …

The number of critical points of the function $f(x)=\{\begin{array}{cc}\left|\frac{\sin x}{x}\right|,& x\ne 0\\ 1,& x=0\end{array}$ in the interval $(-2\pi ,2\pi )$ is equal to:

${\max }_{0\le x\le \pi }\left(16\sin \left(\frac{x}{2}\right){\cos }^3\left(\frac{x}{2}\right)\right)$ is equal to :

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x,y\in R$ and $f^{\prime }(0)=3$. Then the minimum value of the function …

Let the parabola $\mathrm{y}={\mathrm{x}}^2+\mathrm{px}+\mathrm{q}$ passing through the point $(1,-1)$ be such that the distance between its vertex and the x-axis is minimum. Then the value of $p^2+q^2$ is :

Let $\mathrm{f}(\mathrm{x})$ be a polynomial of degree 5, and have extrema at $x=1$ and $x=-1$. If ${\lim }_{x\to 0}\left(\frac{f(x)}{x^3}\right)=-5$, then $\mathrm{f}(2)-\mathrm{f}(-2)$ is equal to :

Let $f(x)={\int }_0^{x^2}\frac{t^2-8t+15}{e^t}dt,x\in \mathbf{R}$. Then the numbers of local maximum and local minimum points of f, respectively, are :

If the set of all values of a , for which the equation $5{\mathrm{x}}^3-15\mathrm{x}-\mathrm{a}=0$ has three distinct real roots, is the interval $(\alpha ,\beta )$, then $\beta -2\alpha$ is equal to

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of $81{\mathrm{cm}}^3/\min$ and the thickness of the ice-cream layer decreases at the rate of …

Let $\mathrm{f}(\mathrm{x})={\int }_0^{\mathrm{x}}\mathrm{t}\left({\mathrm{t}}^2-9\mathrm{t}+20\right)\mathrm{dt},1\le \mathrm{x}\le 5$. If the range of f is $[\alpha ,\beta ]$, then $4(\alpha +\beta )$ equals:

Let $f:[0,3]\to \mathrm{A}$ be defined by
$f(\mathrm{x})=2{\mathrm{x}}^3-15{\mathrm{x}}^2+36\mathrm{x}+7$ and $\mathrm{g}:[0,\infty )\to \mathrm{B}$ be defined by $\mathrm{g}(\mathrm{x})=\frac{{\mathrm{x}}^{2025}}{{\mathrm{x}}^{2025}+1}$. If both the functions ar …

Let $(2,3)$ be the largest open interval in which the function $f(x)=2{\log }_e(x-2)-x^2+ax+1$ is strictly increasing and ( $\mathrm{b},\mathrm{c}$ ) be the largest open interval, in which the function $\mathrm{g}(\mathrm{x})=(\mathrm{x}-1{)}^3(\mathrm{x}+2-\mathrm{a}{)}^2$ …

Consider the region
$R=\left\{(x,y):x\le y\le 9-\frac{11}{3}{x}^2,x\ge 0\right\}$. The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is :

Let $\mathrm{f}:\mathrm{R}\to \mathrm{R}$ be a function defined by $\mathrm{f}(\mathrm{x})=\Vert |\mathrm{x}+2|-2|\mathrm{x}\Vert |$. If m is the number of points of local minima and $n$ is the number of points of local maxima of f , then $\mathrm{m}+\mathrm{n}$ is

If the function $f(x)=2x^3-9a{x}^2+12a^{2}x+1$, where $a>0$, attains its local maximum and local minimum values at $p$ and $q$, respectively, such that $p^2=q$, then $f(3)$ is equal to:

Let $\mathrm{a}>0$. If the function $\mathrm{f}(\mathrm{x})=6{\mathrm{x}}^3-45\mathrm{a}{\mathrm{x}}^2+108{\mathrm{a}}^2\mathrm{x}+1$ attains its local maximum and minimum values at the points ${\mathrm{x}}_1$ and ${\mathrm{x}}_2$ respectively such that …

Let $f:\mathbf{R}\to \mathbf{R}$ be a polynomial function of degree four having extreme values at $\mathrm{x}=4$ and $\mathrm{x}=5$.

If ${\lim }_{\mathrm{x}\to 0}\frac{f(\mathrm{x})}{{\mathrm{x}}^2}=5$, then $f(2)$ is equal to :

Let $\mathrm{x}=-1$ and $\mathrm{x}=2$ be the critical points of the function $\mathrm{f}(\mathrm{x})={\mathrm{x}}^3+{\mathrm{ax}}^2+\mathrm{b}{\log }_{\mathrm{e}}|\mathrm{x}|+1,\mathrm{x}\ne 0$. Let $m$ and M respectively be the absolute minimum and the absolute maximu …

Let the function $\mathrm{f}(\mathrm{x})=\frac{\mathrm{x}}{3}+\frac{3}{\mathrm{x}}+3,\mathrm{x}\ne 0$ be strictly increasing in $\left(-\infty ,{\alpha }_1\right)\cup \left({\alpha }_2,\infty \right)$ and strictly decreasing in …

Let the sum of the maximum and the minimum values of the function $f(x)=\frac{2x^2-3x+8}{2x^2+3x+8}$ be $\frac{m}{n}$, where $\gcd (m,n)=1$. Then $m+n$ is equal to

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 $f$, then ${\alpha }^2+2{\beta }^2$ is equal to

Let $f:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function such that $f(0)=0,f(1)=1,f(2)=-1,f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $(3f^{\prime }{f}^{\prime \prime }+f{f}^{\prime \prime \prime })(x)$ is ____.

For the function $f(x)=\sin x+3x-\frac{2}{\pi }(x^2+x)$, where $x\in [0,\frac{\pi }{2}]$, consider the following two statements: (I) f is increasing in $(0,\frac{\pi }{2})$; (II) $f^{\prime }$ is decreasing in $(0,\frac{\pi }{2})$. Between the above two statements,

Let a variable line of slope $m>0$ passing through the point $(4,-9)$ intersect the coordinate axes at the points $A$ and $B$. The minimum value of the sum of the distances of $A$ and $B$ from the origin is

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

If the function $\mathrm{f}(\mathrm{x})={\left(\frac{1}{\mathrm{x}}\right)}^{2\mathrm{x}},x>0$ attains the maximum value at $\mathrm{x}=\frac{1}{\mathrm{e}}$ then

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

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:

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 $x$ is $[0,\pi ]$.
(S2) $f(x)$ is decreasing in $\left[0,\frac{\pi }{2}\right]$ and increasing in $\left[\frac{\pi }{2},\pi \right]$.

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

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

A variable line L passes through the point $(3,5)$ and intersects the positive coordinate axes at the points A and B. The minimum area of the triangle OAB, where O is the origin, is :

Let the set of all positive values of $\lambda$, for which the point of local minimum of the function
$f(x)=1+x\left({\lambda }^2-x^2\right)$ satisfies $\frac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha ,\beta )$.
Then ${\alpha }^2+{\beta }^2$ is equal to ______.

Let the set of all values of $p$, for which
$f(x)=\left(p^2-6p+8\right)\left({\sin }^{2}2x-{\cos }^{2}2x\right)+2(2-p)x+7$ does not have any critical point, be the interval $(\mathrm{a},\mathrm{b})$. Then 16 ab is equal to

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

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

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

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

Let $\mathrm{g}:\mathrm{R}\to \mathrm{R}$ be a non constant twice differentiable such that $g^{\prime }\left(\frac{1}{2}\right)=g^{\prime }\left(\frac{3}{2}\right)$. If a real valued function f is defined as …

Let $g(x)=3f\left(\frac{x}{3}\right)+f(3-x)$ and $f^{\prime \prime }(x)>0$ for all $x\in (0,3)$. If g is decreasing in ( $0,\alpha$ ) and increasing in ( $\alpha ,3$ ), then $8\alpha$ is

Let for a differentiable function $\mathrm{f}:(0,\infty )\to \mathrm{R}$, $\mathrm{f}(\mathrm{x})-\mathrm{f}(\mathrm{y})\ge {\log }_{\mathrm{e}}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)+\mathrm{x}-\mathrm{y},\forall \mathrm{x},\mathrm{y}\in (0,\infty )$. Then …

Let $S=(-1,\infty )$ and $f:S\to \mathbb{R}$ be defined as $f(x)={\int }_{-1}^x(e^t-1{)}^{11}(2t-1{)}^5(t-2{)}^7(t-3{)}^{12}(2t-10{)}^{61}dt$
Let $p=$ Sum of square of the values of $x$, where $f(x)$ attains local maxima on $S$, and $q=$ Sum of the values of $x$, where …

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

In an A.P., the sixth term $a_6=2$. If the product $a_1{a}_4{a}_5$ is the greatest, then the common difference of the A.P. is equal to

Consider the function $\mathrm{f}:\left[\frac{1}{2},1\right]\to \mathrm{R}$ defined by $f(x)=4\sqrt{2}{x}^3-3\sqrt{2}x-1$. Consider the statements
(I) The curve $\mathrm{y}=\mathrm{f}(\mathrm{x})$ intersects the x-axis exactly at one point
(II) The curve $y=f(x)$ in …