Applications of Derivatives Questions - JEE

52. A sector is to be cut from a circular piece of wire of perimeter 20 metres (i.e., $r + r + r\theta = 20$ metres). The maximum area (in m²) of the sector is:

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

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

Let $f(x) = \begin{cases} -x^3 + a, & 0 \leq x < 1 \\ x, & 1 \leq x \leq 3 \end{cases}$:

The curve $y=f(x)$ satisfies $\dfrac{d^2y}{dx^2}=6x-4$ and has a local minimum value $5$ at $x=1$. $f(0)$ is

Let the function $f(x)=2x^3+(2p-7)x^2+3(2p-9)x-6$ have a maxima for some value of $x<0$ and a minima for some value of $x>0$. Then the set of all values of p is:

If the equation of the normal to the curve $y=\dfrac{x-a}{(x+b)(x-2)}$ at the point $(1,-3)$ is $x-4y=13$, then the value of $a+b$ is equal to ___.

If the functions $f(x)=\dfrac{x^3}{3}+\dfrac{ax^2}{2}+2bx$ and $g(x)=\dfrac{x^3}{3}+bx^2+ax$, $a\neq2b$, have a common extreme point, then $a+2b+7$ is equal to:

If $f(x)=x^2+g'(1)x+g''(2)$ and $g(x)=f(1)x^2+xf'(x)+f''(x)$, then the value of $f(4)-g(4)$ is equal to ___.

Let $f:\mathbb{R}\to\mathbb{R}$ be a differentiable function such that $f'(x)+f(x)=\displaystyle\int_0^2 f(t)\,dt$. If $f(0)=e^{-2}$, then $2f(0)-f(2)$ is equal to ___.

The sum of the absolute maximum and minimum values of the function $f(x)=|x^2-5x+6|-3x+2$ in the interval $[-1,3]$ is equal to:

A triangular park is enclosed on two sides by a fence and on the third side by a straight river bank. The two sides having fence are of same length $x$. The maximum area enclosed by the park is

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)

Given function is $f(x) = \cot^{-1}x + x$; $D_f = \mathbb{R}$. Which of the following is true?

Let $f(x) = \begin{vmatrix} x^3 + \sin x & \cos x & 1 \\ p^2 & p & 0 \\ p^3 & p^2 & p \end{vmatrix}$, where p is a constant. Then $\frac{d^{33}}{dx^{33}}\{f(x)\}$ at $x = 0$ is

If $y = \sin(mx)$, the value of the determinant $\begin{vmatrix} y & y_1 & y_2 \\ y_3 & y_4 & y_5 \\ y_6 & y_7 & y_8 \end{vmatrix}$, where $y_n = \frac{d^n y}{dx^n}$, is

If $f(x) = \dfrac{e^x}{x^2}$, then which of the following is correct?

Find the range of the function $f(x) = \frac{1}{|\sin x|} + \frac{1}{|\cos x|}$

The shortest distance between the line y − x = 1 and the curve x = y2 is

If a b and f(x) = |x − a| + |x − b|, x ∈ ℝ, then the minimum value of f(x) is

The normal to the curve $y(x-2)(x-3) = x + 6$ at the point where the curve intersects the Y-axis passes through the point

Let S be the set of real values of parameter λ for which the equation $f(x) = 2x^3 - 3(2 + \lambda)x^2 + 12\lambda x$ has exactly one local maximum and exactly one local minimum. Then, S is a subset of

The equation of tangent drawn to the curve y2 + 2x3 + 4y − 8 = 0 from the point (1, 2) is given by

Example 54: If the function f(x) = x^3 + 9x^2 - 24x + c has three real and distinct roots N, P and S, then the value of [N] + [P] + [S] are

The greatest of the numbers $1, 2^{1/2}, 3^{1/3}, 4^{1/4}, 5^{1/5}, 6^{1/6}$ and $7^{1/7}$ is:

Let $f(x) = \int_x^{x^3} \frac{dt}{2\ln t}$ for $x > 1$ and $g(x) = \int_1^x (2t^2 - \ln t)f(t)dt$ $(x > 1)$, then:

Let $f(x) = \min\left(\frac{1}{2} - \frac{3x^2}{4}, \frac{5x^2}{4}\right)$ for $0 \leq x \leq 1$. Then the maximum value of $f(x)$ is:

The function $f(x) = \cot^{-1} x + x$ increases in the interval

Given the curve $\sin y = x\sin\left(\dfrac{\pi}{3} + y\right)$, the equation of the normal at $(0, 0)$ is:

The set of values of $p$ for which $f(x) = p^2x - \int 2^{4-x^2}\,dx$ is increasing for all $x \in R$, is:

A wire of length 2 units is cut into two parts which are bent, respectively, to form a square of side $x$ units and a circle of radius $r$ units. If the sum of the areas of the square and the circle so formed is minimum, then

Let $f(x) = x^3 + 6x^2 + ax + 2$. If $(-3, -1)$ is the largest possible interval for which $f(x)$ is a decreasing function, then $a = $

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:

The volume of the largest possible right circular cylinder that can be inscribed in a sphere of radius $= \sqrt{3}$ is

If $k \sin^2 x - \frac{1}{k} \cosec^2 x = 2$, where $x \in \left(0, \frac{\pi}{2}\right)$, then $\cos^2 x - 5 \sin x \cos x - 6 \sin^2 x$ is equal to:

If y = \log x \cdot e^{(\tan x + x^2)}, then \frac{dy}{dx} is equal to

If $p(x)$ be a polynomial of degree three that has a local maximum value 8 at $x = 1$ and a local minimum value 4 at $x = 2$; then $p(0)$ is equal to

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

If $f(x) = x^3 + bx^2 + cx + d$ and $0 \leq b^2 \leq c$, then

If x = −1 and x = 2 are extreme points of \[f(x) = B \log|x| + Cx^2 + x\], then:

Determine the points of maxima and minima of the function $f(x) = \frac{1}{8}\log x - bx + x^2,\, x > 0$ where $b \geq 0$ is a constant.

Consider the two graphs y = 2x and x2 − xy + 2y2 = 28. The absolute value of the tangent of the angle between the two curves at the points where they meet, is ……….

Ex. 13 Statement I: Tangent drawn at the point $(0, 1)$ to the curve $y = x^3 - 3x + 1$ meets the curve thrice at one point only.

If x + y = 2 touches the curve $\frac{x^n}{a^n} + \frac{y^n}{b^n} = 2$ at the point (N, P), then

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

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

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

If the function $f(x)=2x^3-9x^2+12a^2x+1$, $a>0$, 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:

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 $f(x)=1+x(\lambda^2-x^2)$ satisfies $\dfrac{x^2+x+2}{x^2+5x+6}<0$, be $(\alpha,\beta)$. Then $\alpha^2+\beta^2$ is equal to _________.

Applications of Derivatives Questions (514)