Applications of Derivatives Questions - JEE

If $y = Ae^{mx} + Be^{nx}$, then the value of $\frac{d^2y}{dx^2} - (m+n)\frac{dy}{dx} + mny$ is equal to

For the function $y = f(x) = (x - b)x(x - c)e^x$, which of the following holds?

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

If the function $f(x) = (x^2 + ax + 2a)e^x$ is a strictly increasing function in $(-\infty, \infty)$. Then the number of integral values of $a$ is:

If $y + x + y = x = c$, where $c > 0$, then $\frac{dy}{dx}$ has the value equal to

Consider $ f(x) = \tan^{-1}\!\left(\sqrt{\dfrac{1+\sin x}{1-\sin x}}\right), x \in \left(0, \dfrac{\pi}{2}\right) $. A normal to $ y = f(x) $ at $ x = \dfrac{\pi}{6} $ also passes through the point

The normal to the curve $x^2 + 2xy - 3y^2 = 0$ at $(1, 1)$

Given quadratic equation is $ax^2 + bx + c = 0$ Let $f'(x) = ax^2 + bx + c$, so $f(x) = \frac{ax^3}{3} + \frac{bx^2}{2} + cx$. Clearly, $f(0) = 0$ and $f(1) = \frac{1}{6}(2a + 3b + 6c) = 0$. Given that $f(0) = 0 = f(1)$, by Rolle's theorem, $f'(x)$ has at least one root in $(0, 1)$. The root of $ax^2 + bx + c = 0$ lies in:

Let $f(x) = \begin{cases} \frac{1}{2}x^3 - x^2 + 10x + 5, & x \leq 1 \\ 4 - 2^x + \log_2(b^2 - 2), & x > 1 \end{cases}$The set of values of $b$ for which $f(x)$ has greatest value at $x = 1$ is given by

49. Consider the function $f(x) = |x^2 - 7x + 12|(x^2 - 7x + 10)(x^2 - 4x + 3)$. Then Rolle's theorem for $f(x)$ is not applicable to which of the following range?

If $x$ and $y$ are given parametrically, then $\dfrac{dy}{dx} = \dfrac{dy/dt}{dx/dt} = \dfrac{\sin t}{\cos t}$. The length of the tangent is:

The absolute minimum and maximum values of the function $\frac{x^2 - x + 1}{x^2 + x + 1}$ are given by

66. (A) In a circle, $r = 5$ cm and $\Delta r = 0.06$ cm. Find the approximate change in area $\Delta A$ using differentials, where $A = \pi r^2$.

Given functions are f(x) = 5 - |x - 2| and g(x) = |x + 1|, where x \in \mathbb{R}. Find ab where a is the value of x at which maximum of f(x) occurs and b is the value of x at which minimum of g(x) occurs.

Let $y = \sqrt{x + \sqrt{x + \sqrt{x + \cdots}}}$, then $\frac{dy}{dx}$ equals

If $2^x + 2^y = 2^{x+y}$, then $\frac{dy}{dx}$ is equal to

The volume of a right circular cone is given by $V = \frac{1}{3}\pi r^2 h$. When $h = 20$ cm, $\frac{dh}{dt} = -4$ cm/s, $r = 10$ cm and $\frac{dr}{dt} = 2$ cm/s, find $\frac{dV}{dt}$ (in cm³/s).

The minimum value of the function f(x) = x^{3/2} - x^{-3/2} + 4 for all permissible real values of x, 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:

Let f(x) = x² − 2x and g(x) = f(f(x) − 1) + f(5 − f(x)), then

Let $f(x)$ be a differentiable function on $[0, 8]$ such that $f(1) = 3$, $f(2) = 1/2$, $f(3) = 4$, $f(4) = -2$, $f(5) = 6$, $f(6) = 1/3$, $f(7) = -1/4$. Then the minimum number of points of interaction of the curve $y = f'(x)f(x)^2$ and $y = f'(x)f(x)^2$ is $k$, then find $k$.

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:

Suppose, $f(x) = e^{ax} + e^{bx}$, where $a \neq b$ and $f''(x) - 2f'(x) - 15f(x) = 0$ for all $x \in \mathbb{R}$. Then, find $ab$.

Which of the following could be the sketch of graph of $y = \frac{d}{dx}(x \log x)$?

For any $x, y \in R$, $xy > 0$. Then the minimum value of $\dfrac{2x}{y^3} + \dfrac{x^3 y}{3} + \dfrac{4y^2}{9x^4}$ is ______.

Given $f'(x) = x^2 - 2x$ and $f(2) = 0$, the curve $y = f(x)$ has a minimum at $x = 2$. If the point of minimum ordinate is $(a, b)$, find $a + 6b$.

On [1, e], the least and greatest values of f(x) = x2 ln x are m and M respectively, then [M + m] is: (where [ ] denotes greatest integer function)

For $x \in \left(0, \dfrac{5\pi}{2}\right)$, define $f(x) = \int_0^x \sqrt{t} \sin t \, dt$. Then $f$ has

Given $f(x) = x^2 + \dfrac{1}{x^2}$ and $g(x) = x - \dfrac{1}{x}$,$h(x) = \dfrac{f(x)}{g(x)}$. The local maximum and local minimum values of $h(x)$ are respectively:

Let $ f(x) = \dfrac{3x^2 + 9x + 17}{3x^2 + 9x + 7} $ Find the maximum value of $ f(x) $.

Let $f: \mathbb{R} \to \mathbb{R}$ be defined by $f(x) = \begin{cases} k - 2x, & \text{if } x \leq -1 \\ 2x + 3, & \text{if } x > -1 \end{cases}$. If $f$ has a local minimum at $x = -1$, then a possible value of $k$ is

Let $f(x) = 30 - 2x - x^3$. If $f(f(f(x))) > f(f(-x))$, find the highest integral value of $x$ satisfying this inequality.

If $y = \frac{\sec x + \tan x}{\sec x - \tan x}$, then $\frac{dy}{dx}$ equals

The value of $a$ for which the function $f(x) = (4a-3)(x + \log 5) + 2(a-7)\cot\dfrac{x}{2}\sin^2\dfrac{x}{2}$ does not possess critical points 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:

Ex. 69 The function $f(x)$ for $-a \leq x \leq a$ is

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

Applications of Derivatives Questions (514)