Application of Derivatives — JEE Advanced PYQs

Consider the function $f:(0,\infty )\to (-\infty ,\infty )$ given by

Then which one of the following statements is TRUE ?

Let P be the point on the parabola $\mathrm{y}={\mathrm{x}}^2$ such that the slope of the tangent to the parabola at the point P is 4. Let Q be the point in the first quadrant lying on the circle ${\mathrm{x}}^2+{\mathrm{y}}^2=2$ such that the slope of the tangent to the circ …

Let $\mathbb{R}$ denote the set of all real numbers. Define the function $f:\mathbb{R}\to \mathbb{R}$ by

Then which one o …

Let $\mathbb{R}$ denote the set of all real numbers. Let $f:\mathbb{R}\to \mathbb{R}$ be defined by

Then which of the following statements …

Let the function $f:\mathbb{R}\to \mathbb{R}$ be defined by

Then the number of solut …

Let S be the set of all twice differentiable functions f from $\mathbb{R}$ to $\mathbb{R}$ such that $\frac{{\mathrm{d}}^2\mathrm{f}}{{\mathrm{dx}}^2}(\mathrm{x})>0$ for all $\mathrm{x}\in (-1,1)$. For $\mathrm{f}\in \mathrm{S}$, let ${\mathrm{X}}_{\mathrm{f}}$ be the number of p …

Let

Let $g:[0,1]\to \mathbb{R}$ be the function defined by

Then, which of the following statements is/are TRUE?

Let $f:\mathbb{R}\to \mathbb{R}$ be defined by
$f(x)=\frac{x^2-3x-6}{x^2+2x+4}$.
Then which of the following statements is (are) TRUE?

Let $f_1:(0,\infty )\to \mathbb{R}$ and $f_2:(0,\infty )\to \mathbb{R}$ be defined by

and …

Let $f_1:(0,\infty )\to \mathbb{R}$ and $f_2:(0,\infty )\to \mathbb{R}$ be defined by

and …

Let ${\psi }_1:[0,\infty )\to \mathbb{R},{\psi }_2:[0,\infty )\to \mathbb{R},f:[0,\infty )\to \mathbb{R}$ and $\mathrm{g}:[0,\infty )\to \mathbb{R}$ be functions such that …

Consider all rectangles lying in the region

and having one side on the x-axis. The area of the rectangle which has the maximum perimeter among all such rec …

For a polynomial g(x) with real coefficient, let $m_g$ denote the number of distinct real roots of g(x). Suppose S is the set of polynomials with real coefficient defined by …

Let the function $f:(0,\pi )\to \mathbb{R}$ be defined by

Suppose the function $f$ has a local minimum at $\theta$ precisely when $\theta \in \{{\lambda }_1\pi ,\dots ,{\lambda }_r\pi \}$, w …

Let $f:\mathbb{R}\to \mathbb{R}$ be given by …

Let $\mathrm{f}:\mathbb{R}\to \mathbb{R}$ be given by $\mathrm{f}(\mathrm{x})=(\mathrm{x}-1)(\mathrm{x}-2)(\mathrm{x}-5)$. Define $\mathrm{F}(\mathrm{x})={\int }_0^{\mathrm{x}}\mathrm{f}(\mathrm{t})\mathrm{dt},\mathrm{x}>0$. Then which of the following options i …

Let $\mathrm{f}(\mathrm{x})=\frac{\sin \pi \mathrm{x}}{{\mathrm{x}}^2},\mathrm{x}>0$
Let ${\mathrm{x}}_1<{\mathrm{x}}_2<{\mathrm{x}}_3<\dots <{\mathrm{x}}_{\mathrm{n}}<\dots$ be all the points of local maximum of f and …

For every twice differentiable function $f:\mathbb{R}\to [-2,2]$ with $(f(0){)}^2+{\left(f^{\prime }(0)\right)}^2=85$, which of the following statement(s) is (are) TRUE?

Let $f(x)=x+{\log }_{\mathrm{e}}x-x{\log }_{\mathrm{e}}x,x\in (0,\infty )$.

• Column 1 contains information about zeros of $f(x),f^{\prime }(x)$ and $f^{\prime \prime }(x)$.
• Column 2 contains information about the limiting behavior of $f(x),f^{\prime }(x)$ and …

Let $f(x)=x+{\log }_{\mathrm{e}}x-x{\log }_{\mathrm{e}}x,x\in (0,\infty )$.

• Column 1 contains information about zeros of $f(x),f^{\prime }(x)$ and $f^{\prime \prime }(x)$.
• Column 2 contains information about the limiting behavior of $f(x),f^{\prime }(x)$ and …

If $f:\mathbb{R}\to \mathbb{R}$ is a twice differentiable function such that $f^{\prime \prime }(x)>0$ for all $x\in \mathbb{R}$, and $f\left(\frac{1}{2}\right)=\frac{1}{2},f(1)=1$, then

If $f:\mathbb{R}\to \mathbb{R}$ is a differentiable function such that $f^{\prime }(x)>2f(x)$ for all $x\in \mathbb{R}$, and $f(0)=1$, then

The total number of distinct $x\in [0,1]$ for which ${\int }_0^x\frac{t^2}{1+t^4}dt=2x-1$ is

Let $f:\mathbb{R}\to (0,\infty )$ and $g:\mathbb{R}\to \mathbb{R}$ be twice differentiable functions such that $f^{\prime \prime }$ and $g^{\prime \prime }$ are continuous functions on $\mathbb{R}$. Suppose …

Let $f,g:[-1,2]\to \mathbb{R}$ be continuous functions which are twice differentiable on the interval $(-1,2)$. Let the values of $f$ and $g$ at the points $-1,0$ and 2 be as given in the following table:
| | $x=-1$ | $x=0$ | $x=2$ || --- | --- | --- | …

PARAGRAPH 2
Let $F:\mathbb{R}\to \mathbb{R}$ be a thrice differentiable function. Suppose that $F(1)=0,F(3)=-4$ and $F^{\prime }(x)<0$ for all $x\in (1/2,3)$. Let $f(x)=xF(x)$ for all $x\in \mathbb{R}$.

The correct statement(s) is(are)

A cylindrical container is to be made from certain solid material with the following constraints: It has a fixed inner volume of $V{\mathrm{mm}}^3$, has a 2 mm thick solid wall and is open at the top. The bottom of the container is a solid circular disc of thickness 2 mm and is …

Let $a\in \mathbb{R}$ and let $f:\mathbb{R}\to \mathbb{R}$ be given by

Then

Let $f:\left[\frac{1}{2},1\right]\to \mathbb{R}$ (the set of all real numbers) be a positive, non-constant and differentiable function such that $f^{\prime }(x)<2f(x)$ and $f\left(\frac{1}{2}\right)=1$. Then the value of ${\int }_{1/2}^{1}f(x)dx$ lies in the i …

The number of points in $(-\infty ,\infty )$, for which $x^2-x\sin x-\cos x=0$, is

Let $f(x)=x\sin \pi x,x>0$. Then for all natural numbers $n,f^{\prime }(x)$ vanishes at

A rectangular sheet of fixed perimeter with sides having their lengths in the ratio 8 : 15 is converted into an open rectangular box by folding after removing squares of equal area from all four corners. If the total area of removed squares is 100, the resulting box has maximum v …

The function $f(x)=2|x|+|x+2|-||x+2|-2|x||$ has a local minimum or a local maximum at $x=$

Let $f:[0,1]\to \mathbb{R}$ (the set of all real numbers) be a function. Suppose the function $f$ is twice differentiable, $f(0)=f(1)=0$ and satisfies $f^{\prime \prime }(x)-2f^{\prime }(x)+f(x)\ge e^x,x\in [0,1]$.

If the function ${\mathrm{e}}^{-x}f(x)$ assumes its …

Let $f(x)=(1-x{)}^2{\sin }^{2}x+x^2$ for all $x\in \mathbb{R}$, and let $g(x)={\int }_1^x\left(\frac{2(t-1)}{t+1}-\ln t\right)f(t)dt$ for all $x\in (1,\infty )$.

Which of the following is true?

If $f(x)={\int }_0^x{e}^{t^2}(t-2)(t-3)dt$ for all $x\in (0,\infty )$, then

The function $f:[0,3]\to [1,29]$, defined by $f(x)=2x^3-15x^2+36x+1$, is

Let $f:\mathbb{R}\to \mathbb{R}$ be defined as $f(x)=|x|+\left|x^2-1\right|$. The total number of points at which $f$ attains either a local maximum or a local minimum is

Let $p(x)$ be a real polynomial of least degree which has a local maximum at $x=1$ and a local minimum at $x=3$. If $p(1)=6$ and $p(3)=2$, then $p^{\prime }(0)$ is

Let f be a function defined on $\mathbf{R}$ (the set of all real numbers) such that ${\mathrm{f}}^{\prime }(\mathrm{x})=2010(\mathrm{x}-2009)(\mathrm{x}-2010{)}^2(\mathrm{x}-2011{)}^3(\mathrm{x}-2012{)}^4$, for all $\mathrm{x}\in \mathbf{R}$. If $g$ is a function defined on …

Consider the polynomial

Let s be the sum of all distinct real roots of $\mathrm{f}(\mathrm{x})$ and let $\mathrm{t}=|\mathrm{s}|$.

The function $f^{\prime }(x)$ is

Let $f,g$ and $h$ be real-valued functions defined on the interval $[0,1]$ by $f(x)=e^{x^2}+e^{-x^2},g(x)=x{e}^{x^2}+e^{-x^2}$ and $h(x)=x^2{e}^{x^2}+e^{-x^2}$. If a, b and c denote, respectively, the absolute maximum of f, g and h on [0, 1], then

For the function

The maximum value of the function $f(x)=2x^3-15x^2+36x-48$ on the set $A=\left\{x\mid x^2+20\le 9x\right\}$ is

The total number of local maxima and local minima of the function

is

Let $f(x)$ be a non-constant twice differentiable function defined on $(-\infty ,\infty )$ such that $f(x)=f(1-x)$ and $f^{\prime }\left(\frac{1}{4}\right)=0$. Then,

The tangent to the curve $y=e^x$ drawn at the point $(c,e^c)$ intersects the line joining the points $(c-1,e^{c-1})$ and $(c+1,e^{c+1})$

Let $f(x)=2+\cos x$ for all real $x$.

STATEMENT-1 : For each real $t$, there exists a point $c$ in $[t,t+\pi ]$ such that $f^{\prime }(c)=0$.

because

STATEMENT-2 : $f(t)=f(t+2\pi )$ for each real $t$.

If a continuous function $f$ defined on the real line $\mathbf{R}$, assumes positive and negative values in $\mathbf{R}$ then the equation $f(x)=0$ has a root in $\mathbf{R}$. For example, if it is known that a continuous function $f$ on $\mathbf{R}$ is positive at …

If a continuous function $f$ defined on the real line $\mathbf{R}$, assumes positive and negative values in $\mathbf{R}$ then the equation $f(x)=0$ has a root in $\mathbf{R}$. For example, if it is known that a continuous function $f$ on $\mathbf{R}$ is positive at …