Differentiation and Applications of Derivatives — JEE Main PYQs

Let $f(x)$ 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 $f(2) - f(-2)$ is equal to :

If $y = \tan^{-1}\left(\frac{3\cos x - 4\sin x}{4\cos x + 3\sin x}\right) + 2\tan^{-1}\left(\frac{x}{1+\sqrt{1-x^2}}\right)$, then $\frac{dy}{dx}$ at $x = \frac{\sqrt{3}}{2}$ is equal to:

Let $f$ be a real polynomial of degree $n$ such that $f(x) = f'(x)\, f''(x)$, for all $x \in \mathbb{R}$. If $f(0) = 0$, then $36\left(f'(2) + f''(2) + \int_0^2 f(x)\,dx\right)$ is equal to:

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

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

Let $f(x)$ and $g(x)$ be twice differentiable functions satisfying $f''(x) = g''(x)$ for all $x \in \mathbf{R}$, $f'(1) = 2g'(1) = 4$ and $g(2) = 3f(2) = 9$. Then $f(25) - g(25)$ is equal to :

If the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$, 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 $f: \mathbf{R} \to \mathbf{R}$ be a twice differentiable function such that $(\sin x \cos y)(f(2x + 2y) - f(2x - 2y)) = (\cos x \sin y)(f(2x + 2y) + f(2x - 2y))$, for all $x, y \in \mathbf{R}$. If $f'(0) = \frac{1}{2}$, then the value of $24 f''\left(\frac{5\pi}{3}\right)$ is :

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

Let $\mathrm{a}>0$. If the function $f(x)=6x^3-45\mathrm{a}x^2+108\mathrm{a}^2x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1x_2=54$, then $\mathrm{a}+x_1+x_2$ is equal to :

Let $x = -1$ and $x = 2$ be the critical points of the function $f(x) = x^3 + ax^2 + b \log_e |x| + 1$, $x \neq 0$. Let $m$ and $M$ respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2, -\frac{1}{2}\right]$. Then $|M + m|$ is equal to (Take $\log_e 2 = 0.7$) :

Let $f : \mathbf{R} \to \mathbf{R}$ be a polynomial function of degree four having extreme values at $x = 4$ and $x = 5$. If $\lim\limits_{x \to 0} \dfrac{f(x)}{x^2} = 5$, then $f(2)$ is equal to :

Let the function $f(x) = \frac{x}{3} + \frac{3}{x} + 3$, $x \neq 0$ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $\sum\limits_{i=1}^{5} \alpha_i^2$ is equal to

Let $f(x) = \displaystyle\int_{0}^{x^2} \dfrac{t^2 - 8t + 15}{e^t} \, dt$, $x \in \mathbb{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

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 $81\ \mathrm{cm}^3/\text{min}$ and the thickness of the ice-cream layer decreases at the rate of $\dfrac{1}{4\pi}$ cm/min. The surface area (in $\mathrm{cm}^2$) of the chocolate ball (without the ice-cream layer) is :

Consider the region $R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3}x^2, x \geq 0\right\}$.

The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R, is:

Let $(2, 3)$ be the largest open interval in which the function $f(x) = 2\log_e(x - 2) - x^2 + \mathrm{a}x + 1$ is strictly increasing and $(\mathrm{b}, \mathrm{c})$ be the largest open interval, in which the function $\mathrm{g}(x) = (x-1)^3(x + 2 - \mathrm{a})^2$ is strictly decreasing. Then $100(\mathrm{a} + \mathrm{b} - \mathrm{c})$ is equal to :

The sum of all local minimum values of the function $$f(x) = \begin{cases} 1 - 2x, & x < -1 \\ \frac{1}{3}(7 + 2|x|), & -1 \le x \le 2 \\ \frac{11}{18}(x-4)(x-5), & x > 2 \end{cases}$$ is

Let $f(x)=x^{5}+2e^{x/4}$ for all $x \in \mathbf{R}$. Consider a function $g(x)$ such that $(g \circ f)(x)=x$ for all $x \in \mathbf{R}$. Then the value of $8g'(2)$ is :

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 $(3f'f'' + ff''')(x)$ is ___________

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

(I) $f$ is increasing in $\left(0, \frac{\pi}{2}\right)$.

(II) $f'$ is decreasing in $\left(0, \frac{\pi}{2}\right)$.

Between the above two statements,

Let $f(x) = x^5 + 2x^3 + 3x + 1$, $x \in \mathbb{R}$, and $g(x)$ be a function such that $g(f(x)) = x$ for all $x \in \mathbb{R}$. Then $\dfrac{g(7)}{g'(7)}$ is equal to :

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 rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then $(a+b)^2$ is equal to :

If $y(\theta) = \dfrac{2\cos\theta + \cos 2\theta}{\cos 3\theta + 4\cos 2\theta + 5\cos\theta + 2}$, then at $\theta = \dfrac{\pi}{2}$, $y'' + y' + y$ is equal to :

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

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

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

Suppose for a differentiable function $h$, $h(0)=0$, $h(1)=1$ and $h'(0)=h'(1)=2$. If $g(x)=h(\mathrm{e}^x)\mathrm{e}^{h(x)}$, then $g'(0)$ is equal to :

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 $[0,\pi]$.

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

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

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

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

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40$, $f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to :

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 $\frac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha, \beta)$. Then $\alpha^2+\beta^2$ is equal to ________.

If $\log_e y = 3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\frac{1}{2}$ 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 $(a, b)$. Then $16ab$ is equal to ________.

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

If $y=\frac{(\sqrt{x}+1)(x^2-\sqrt{x})}{x\sqrt{x}+x+\sqrt{x}}+\frac{1}{15}(3\cos^2 x - 5)\cos^3 x$, then $96\, y'\!\left(\frac{\pi}{6}\right)$ is equal to ________.

Let for a differentiable function $f : (0, \infty) \to \mathbf{R}$, $f(x) - f(y) \geq \log_e\left(\dfrac{x}{y}\right) + x - y,\ \forall\ x, y \in (0, \infty)$.

Then $\sum\limits_{n=1}^{20} f'\left(\dfrac{1}{n^2}\right)$ is equal to ________.

Let $f(x) = x^3 + x^2 f'(1) + x f''(2) + f'''(3)$, $x \in \mathbf{R}$. Then $f'(10)$ is equal to ________.

Suppose $f(x)=\dfrac{\left(2^x + 2^{-x}\right)\tan x\sqrt{\tan^{-1}\left(x^2 - x + 1\right)}}{\left(7x^2 + 3x + 1\right)^3}$. Then the value of $f'(0)$ is equal to

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

Consider the statements

(I) The curve $y=f(x)$ intersects the $x$-axis exactly at one point.

(II) The curve $y=f(x)$ intersects the $x$-axis at $x=\cos\dfrac{\pi}{12}$.

Then

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 $y=f'(x)$ intersect the $x$-axis, then the value of $m + n$ is __________.

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

Let $f: \mathbb{R}-\{0\} \to \mathbb{R}$ be a function satisfying $f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$ for all $x, y,\ f(y) \neq 0$. If $f'(1)=2024$, then

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 of $M-m$ is

Let $S=(-1, \infty)$ and $f: S \rightarrow \mathbb{R}$ be defined as

$$f(x)=\int_{-1}^{x}\left(e^t-1\right)^{11}(2t-1)^5(t-2)^7(t-3)^{12}(2t-10)^{61}\,dt,$$

Let p = Sum of squares of the values of $x$, where $f(x)$ attains local maxima on S, and q = Sum of the values of $x$, where $f(x)$ attains local minima on S. Then, the value of $\mathrm{p}^2+2\mathrm{q}$ is ________