Applications of Derivatives Questions - JEE

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 _________.

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:

Applications of Derivatives Questions (514)