Applications of Derivatives Questions - JEE

Given: $f(x) = 2x^3 - 9x^2 + 12x + 5$ on $[0, 3]$ Find $M - m$ where $M$ is the maximum value and $m$ is the minimum value of $f(x)$ on $[0,3]$.

The set of all values of b for which the function f(x) = (b^2 + 3b + 2)(\cos^2 x + \sin^2 x) + (b + 1)x + \sin 2 does not possess stationary points is

Let $f(x) = \int_0^x (\sin t + \cot t)(e^t - 2)(t - 1)^3(t - 2)^5 dt$ for $0 , then the number of points where \(f(x)$ assumes local maximum value is

For the curve $y = 3\sin\theta\cos\theta$, $x = e^\theta\sin\theta$, $0 \le \theta \le \pi$, the tangent is parallel to x-axis when $\theta$ is

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

If f(x) = \begin{cases} 3 + |x − k|, & \text{for } x > k \\ \sin(x − k)/(x − k), & \text{for } x k \\ a, & \text{for } x = k \end{cases} has minimum at x = k, then

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

Let $f:[2,4]\to\mathbb{R}$ be a differentiable function such that $(x\log_e x)f'(x)+(\log_e x)f(x)+f(x)\geq 1,\ x\in[2,4]$ with $f(2)=\dfrac{1}{2}$ and $f(4)=\dfrac{1}{2}$. Consider the following two statements: (A) $f(x)\leq 1$, for all $x\in[2,4]$; (B) $f(x)\geq\dfrac{1}{8}$, for all $x\in[2,4]$. Then,

If e^y(x+1) = 1, then \frac{d^2y}{dx^2} is equal to

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

Ex. 67 The function $f(x)$ is

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

Let for a differentiable function $f:(0,\infty)\to\mathbb{R}$, $f(x)-f(y)\geq\log_e\left(\dfrac{x}{y}\right)+x-y$, $\forall x,y\in(0,\infty)$. Then $\displaystyle\sum_{n=1}^{20}f'\left(\dfrac{1}{n^2}\right)$ is equal to

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

The minimum value of the function $y = |x+2| + |x-2| + |x-5| + |x-8| + |x-10|$ is:

If the graph of the function $f(x) = 3x^4 + 2x^3 + ax^2 - x + 2$ is concave up for all real values of $x$, then the value of $a$ is

The least value of $\alpha \in R$ for which $4\alpha x^2 + \dfrac{1}{x} \ge 1$, for all $x > 0$, is

If the total maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\!\sin 2x},\ x\in\left(0,\pi\right)$, is $k$, then $k^8_e + k^8_{e^2} + k^8_{e^5}$ is equal to

Let $f(x)=4\cos^3 x+3\sqrt{3}\cos^2 x-10$. The number of points of local maxima of $f$ in the interval $(0,2\pi)$ 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 angle between the curves $x^3 - 3xy^2 + 2 = 0$ and $3x^2 y - y^3 - 2 = 0$ is:

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

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

The maximum value of $f(x) = \cos x(1 + \cos x)$ is greater than its minimum value by:

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 \(0

A cone is inscribed in a sphere of radius $r$. The curved surface area of the cone is maximum when $\sin\alpha = \dfrac{1}{3}$, where $\alpha$ is the semi-vertical angle. The maximum curved surface area of the cone is:

Let $f(x)$ be a cubic polynomial on $\mathbb{R}$ which increases in the interval $(-\infty, 0) \cup (1, \infty)$ and decreases in the interval $(0, 1)$. If $f'(2) = 6$ and $f(2) = 2$, then the value of $\tan^{-1}(f(1)) + \tan^{-1}\!\left(f\!\left(\dfrac{3}{2}\right)\right) + \tan^{-1}(f(0))$ is equal to:

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

Let f(x) and g(x) be two functions which are defined and differentiable for all x \geq x_0. If f(x_0) = g(x_0) and f'(x) > g'(x) for all x > x_0, then

We have $\dfrac{d(p(t))}{dt} = \dfrac{1}{2}p(t) - 450$. At $t = 0$, $p(0) = 50$. Find the time $t$ when $P(t) = 0$.

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

Given $y = \dfrac{x}{x^2 - 3}$. The curve passes through the point $(\alpha, \beta)$ and the tangent at $(\alpha, \beta)$ is parallel to the line $2x + 6y - 11 = 0$. Then $|6\alpha + 2\beta|$ equals:

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 cm/s, then the rate (in cm/sec) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is ________ (up to three decimal places).

The equation of family of curves for which the length of normal at any point P is equal to the distance of P from origin, is

Ex. 14: If \(D = 4(a^2 - 3b)

101. Slope of tangent to the curve $y = 2e^x \sin\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)\cos\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)$ where $0 \leq x \leq 2\pi$, is minimum at $x$ is equal to:

The sum of all local minimum values of the function ⎧ 1 - 2x, x < -1 ⎪ ⎪ 1 f (x) = ⎨ (7 + 2|x|), -1 \le x \le 2 3 ⎪ ⎩ ⎪ 11 (x - 4)(x - 5), x > 2 18 is

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

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\displaystyle\int_0^x e^{(x-t)}f(t)\,\mathrm{d}t$, $x\in\mathbf{R}$ and let $g(x)=\displaystyle\int_0^x (f(t)+2)^{15}(t-4)^6(t+12)^{17}\,\mathrm{d}t$, $x\in\mathbf{R}$. If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to _____

Let $f(x)=\dfrac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}$, $x\in[0,\pi]-\left\{\dfrac{\pi}{4}\right\}$, then $f\!\left(\dfrac{7\pi}{12}\right)f''\!\left(\dfrac{7\pi}{12}\right)$ is equal to

The tangent to the curve $y = xe^{x^2}$ at the point $(1, e)$, also passes through the point:

Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $P(1,3)$ on it meet the $y$-axis at $A$. Let the line passing through $P$ and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at $B$. If $B$ lies on the line $2x-3y=8$, then $(AB)^2$ is equal to _______.

A cone has slant height $l = 3$ m. The maximum volume (in cu. m) of the cone is:

When the given equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0, 1]$, then $f'(x)$ will change sign. But $f'(x) = 6x^2 + 3 > 0$, for all values of $x \in \mathbb{R}$. The number of values of $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0,1]$ is:

For the parametric curve x = x(t), y = y(t), find \frac{dy}{dx} at the point corresponding to t = 2, given that at this point \frac{dy}{dt} = 4t - 2 and \frac{dx}{dt} = 2t + 3.

Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha,\alpha+1)$ in the first quadrant is _______.

A point $ x = x_1 $ in the domain of $ f $ is said to be a stationary point if $ f'(x_1) = 0 $. Which of the following statements is/are true?(1) Every local maximum or minimum of $ f $ is a stationary point.(2) Every stationary point of $ f $ is a local maximum or minimum.(3) If $ f $ has an absolute maximum on an open interval $(a,b)$, it must occur at a stationary point.(4) $ f $ is continuous on $(a,b)$ implies $ f $ has an absolute extremum on $(a,b)$.

Applications of Derivatives Questions (514)