Applications of Derivatives Questions - JEE

976. Find the least value of the function $y = x^2 e^{-x} + 4 - \sqrt{4 - x^2}$.

Consider the region $R=\left\{(x,y)\in\mathbb{R}^2: 0\leq x\leq\dfrac{\pi}{3},\; 0\leq y\leq 4\cos\!\left(3x-\dfrac{\pi}{2}\right)\right\}$. A rectangle is inscribed in $R$ with one side on the $x$-axis. Let $A_0$ be the area of the rectangle that has the maximum perimeter among all such rectangles. Then the value of $\dfrac{9A_0\sin^{-1}(1/6)}{4\sin(3x_0)}$ (where $x_0$ is the $x$-coordinate of the optimal rectangle's corner) is:

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

Let $f(x)$ be a cubic polynomial such that its local maximum is at $(0,1)$ and local minimum at $(1,-2)$, and $f(-1)0$. The value of $\sin^{-1}(\cos[\alpha])$ may be equal to (where $\alpha$ is a root of $f(x)=0$, and $[\cdot]$ is the greatest integer function)

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

The approximate change in the volume of a cube of side x m caused by increasing the side by 3% is:

If $y = x^{x^2}$, then $\frac{dy}{dx}$ equals

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

Given curve $y = x^3 + ax - b$. The slope of the tangent to this curve at $(1, -5)$ is perpendicular to the line $x - y + 5 = 0$. Find the values of $a$ and $b$.

Let $f(x)$ be a cubic polynomial such that its local maximum is at $(0,1)$ and local minimum at $(1,-2)$, and $f(-1)<0$, $f(2)>0$. The value of $\sin^{-1}(\cos[\alpha])$ may be equal to (where $\alpha$ is a root of $f(x)=0$, and $[\cdot]$ is the greatest integer function)

Let $P = x^3 - \frac{1}{x^3}, Q = x - \frac{1}{x}$ and $'a'$ is the minimum value of $\frac{P}{Q^2}$. Then the value of $[a]$ is ________. (where $[x] = $ the greatest integer $\leq x$).

The smallest positive integral value of $p$ for which the function $f(x) = 6px - p\sin 4x - 5x - \sin 3x$ is monotonic increasing and has no critical points on $R$ is:

A polynomial function $P(x)$ of degree 5 with leading coefficient one, increases in the interval $(-\infty, 1)$ and $(3, \infty)$ and decreases in the interval $(1, 3)$. Given that $P(0) = 4$ and $P'(2) = 0$. Find the value $P'(6)$.

A particle just clears a wall of height b at distance a and strikes the ground at a distance c from the point of projection. The angle of projection is

Using mean value theorem, if $f(x) = \log x$ on $[1, 3]$, find the value of $c$ such that $f'(c) = \frac{f(3) - f(1)}{3 - 1}$. What is $c$?

Find the value of |a| for which the area of triangle included between the coordinate axes and any tangent to the curve x^a y = l^a is constant (where l is constant).

Let $f(x)=e^x$ and $g(x)=e^{-x}$. Then, by LMVT, for $[0,1]$:

A water tank has the shape of an inverted circular cone with axis vertical. Semi-vertical angle is $\tan^{-1}(0.5)$. Water is poured at 5 cubic metres per hour. Rate at which water level is rising when depth is 4m is (Take $\pi=22/7$)

The given function is $f(x) = (x+1)^{1/3} - (x-1)^{1/2}$; $x \in [0,1]$. What is the greatest value of $f(x)$ on $[0,1]$?

Let $f:(0,\infty) \to R$ be a differentiable function satisfying $f(x) + e^{f(x)} = \dfrac{2}{x} - \ln x - 1$. Find the number of integers in the range of $x$ satisfying the inequality $f(2x^2+1) - f(x^2+5) \ge f(1),\, x > 0$.

A line through $(2,3)$ intersects the curve $y^2=4x$ at $A$ and $B$. The angle subtended by arc $AB$ at the vertex $(0,0)$ is $90°$. Then the line $AB$ is

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

Two ships $A$ and $B$ start from points $A$ and $B$ respectively. Ship $A$ travels $20t$ and ship $B$ travels $30t$ in time $t$. The angle between their directions is $120°$. The rate of change of the distance between $A'$ and $B'$ at $t = 0$ 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 common tangent is drawn to the parabolas $y = x^2 - x + 1$ and $y = x^2 - 3x + 1$. If the slope of the common tangent is $m$, find $|m|$.

Let $f$ be a differentiable function satisfy $x^2 f'(x) + 2x f(x) = e^x$ and $f(2) = \dfrac{e^2}{4}$, then:

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:

The number of critical points of \[f(x) = \max\{\sin x, \cos x\}, \quad x \in (-2\pi, 2\pi)\] is

A window of fixed perimeter (including the base of the arch) is in the form of a rectangle surmounted by a semicircle. The semicircular portion is fitted with coloured glass while the rectangular part is fitted with clear glass. The clear glass transmits three times as much light per square metre as the coloured glass does. What is the ratio of the sides of the rectangle so that the window transmits maximum light?

The value of function $$f(x) = 1 + x + \int_1^x (\ln^2 t + 2 \ln t) \, dt$$ where $$f'(x)$$ vanishes is:

The differential equation $y^3 y'' = -1$ with $y'y'' = \frac{-y'}{y^3}$, is solved with conditions $x=1,\ y=1$ and $y'(1)=0$. The function $f(x) = y = \sqrt{2x - x^2}$ is defined for $y \geq 0,\ x \in D_f$. What is the maximum value of $f(x)$?

If $x, y \in R^+$ satisfying $x + y = 3$, then the maximum value of $x^2 y$ is ______.

Find the range of values of a for which the function $f(x) = x^3 + (a+2)x^2 + 3ax + 5$ is monotonic in $\mathbb{R}$. Hence, find the set of values of a for which $f(x)$ is invertible.

The equation of normal to x + y = xy, where it intersects the X-axis, is given by

Let $f(x) = (x^2 + ax + 2a)e^x$. If $f(x)$ is an increasing function for all $x \in \mathbb{R}$, find the number of integral values of $a$.

A water tank has the shape of an inverted right circular cone, whose semi-vertical angle is $\tan^{-1}(1/2)$. Water is poured into it at a constant rate of 5 cubic meter per minute. Then the rate (in m/min.), at which the level of water is rising at the instant when the depth of water in the tank is 10 m, is ________ (up to four decimal places).

A stone is dropped into a quiet lake and waves move in circles at the speed of 5 cm/sec. At that instant, when the radius of circular wave is 8 cm, the rate of increase of enclosed area is:

A company makes 3 model of calculators: A, B and C at factory I and factory II. The company has ordered for at least 6400 calculators of model A, 4000 calculators of model B and 4800 calculators of model C. At factory I, 50 calculators of model A, 50 of model B and 30 of model C are made every day; at factory II, 40 calculators of model A, 20 of model B and 40 of model C are made everyday. It costs ₹12000 and ₹15000 each day to operate factory I and II, respectively. Find the number of days each factory should operate to minimise the operating costs and still meet the demand.

The equation of the tangent to the curve $y = x - \frac{8}{x^2}$ which is parallel to the x-axis is

Let $l$ be the line through $(0, 0)$ and tangent to the curve $y = x^3 + x + 16$. Then the slope of $l$ is equal to:

Let \(f(x) = -x^4 + \dfrac{b^3 - b^2 + b - 1}{b^2 + 3b + 2},\quad 0 \leq x

Let $f(x) = 1 + \int_0^1 (xe^y + ye^x)f(y)\,dy$ where $x$ and $y$ are independent variables.If the acute angle of intersection of the curves $\frac{x}{2} + \frac{y}{3} + \frac{1}{3} = 0$ and $y = f(x)$ is $\theta$, then $\tan\theta$ equals to:

Example 53: If f(x) is a polynomial of degree 5 with real coefficients such that f(|x|) = 0 has 8 real roots, then f(x) = 0 has

Function $f(x) = 2x^2 - \log|x|, x \neq 0$ monotonically increases in

The function $x^4 - 4x + 1$ will have

Let $f : [1, \infty) \to \mathbb{R}$ and $f(x) = x-1$. Then which statement is true?(A) f(x) is an increasing function(B) $\lim_{x \to \infty} f(x) = \infty$(C) f(x) has a maxima at $x = e$(D) f(x) is a decreasing function

Let $S$ be the set of all values of $x$ for which the tangent to the curve $y = f(x) = x^3 - x^2 - 2x$ at $(x, y)$ is parallel to the line segment joining the points $(1, f(1))$ and $(-1, f(-1))$, then $S$ is equal to:

Let $f(x) = \begin{cases} x^2 - 2|x| + a, & x \leq 1 \\ 6 + x, & x > 1 \end{cases}$, then number of positive integral value(s) of $a$ for which $f(x)$ has local minima at $x = 1$, is/are:

Applications of Derivatives Questions (514)