Applications of Derivatives Questions - JEE

The two curves $C_1: x^3 - 3xy^2 + 2 = 0$ and $C_2: 3x^2y - y^3 - 2 = 0$

The corner points of the feasible region determined by the system of linear constraints are $(0, 0)$, $(0, 40)$, $(20, 40)$, $(60, 20)$, $(60, 0)$. The objective function is $z = 4x + 3y$.Compare the quantity in Column A and Column B:Column AColumn BMaximum of $z$325

Let \(0

The function $f(x)=2x+3(x)^{2/3}$, $x\in\mathbb{R}$, has

Let $f(x) = x^2 + \dfrac{1}{x^2}$ and $g(x) = x - \dfrac{1}{x}$, $x \in \mathbb{R} - \{-1, 0, 1\}$. If $h(x) = \dfrac{f(x)}{g(x)}$, then the local minimum value of $h(x)$ is

The set of all $a\in\mathbb{R}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is

Let $f(x) = x^3 + x + 1$ and $g(x)$ be its inverse, then equation of tangent to $y = g(x)$ at $x = 3$ is:

$\max_{0\leq x\leq\pi}\left\{x-2\sin x\cos x+\dfrac{1}{3}\sin 3x\right\}=$

If $f(x) = \dfrac{2}{\sin x}$ and $g(x) = \dfrac{2}{\tan x}$ where \(0

The function $f(x) = \sin^4 x + \cos^4 x$ is increasing, if

Towns A and B are situated on the same side of a straight road at distances $a$ and $b$ respectively from it. Perpendiculars drawn from A and B meet the road at the points C and D respectively. The distance between C and D is $c$. A hospital is to be built at a point P on the road between C and D such that the distance APB is minimum. Find the position of P.

Let $P$ be a point on the curve $c_1: y = 2 - x^2$ and $Q$ be a point on the curve $c_2: xy = 9$, both $P$ and $Q$ in the first quadrant. If $d$ denotes the minimum distance between $P$ and $Q$, then $d^2$ is ………

Ex. 16: Statement I If differentiable function $f(x)$ satisfies the relation $f(x) + f(x+2) = 0$, $\forall x \in \mathbb{R}$, and if $\frac{1}{d}\frac{d}{dx}f(x)\bigg|_{x=a} = 6$, then $\frac{1}{d}\frac{d}{dx}f(x)\bigg|_{x=a+4000} = 6$Statement II $f(x)$ is a periodic function with period 4.

Ex. 18: If $D = 4(a^2 - 3b) = 0$, then $f(x) = x^3 + ax^2 + bx + c$

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

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:

The maximum value of the function $f(x) = 2x^3 - 15x^2 + 36x - 48$ on the set $A = \{x \mid x^2 + 20 \leq 9x\}$ is:

If $2x^y+3y^x=20$, then $\dfrac{dy}{dx}$ at $(2,2)$ is equal to:

The figure shows $y = P(x) = ax^5 + bx^4 + cx^3 + dx^2 + ex + f$. If $P''(x)$ has real roots $\alpha, \beta, \gamma$, then find $[\alpha] + [\beta] + [\gamma]$, where $[\cdot]$ denotes the greatest integer function.

If \[ f(x) = \begin{cases} -e^{-x} + k, & x \leq 0 \\ e^x + 1, & 0

The point of extremum of $f(x) = \int_0^x (t-2)^2(t-1)\,dt$ is a

If y = (1 + x)(1 + x^2)(1 + x^4) \cdots (1 + x^{2^n}), then the value of \frac{dy}{dx} at x = 0 is

Let the maximum and minimum values of $\left(\sqrt{8x-x^2-12}-4\right)^2+(x-7)^2$, $x\in\mathbb{R}$, be $M$ and $m$ respectively. Then $M^2-m^2$ is equal to _________.

If $x, y, z$ are positive real numbers, such that $x + y + z = 1$. If the minimum value of $\left(1 + \dfrac{1}{x}\right)\left(1 + \dfrac{1}{y}\right)\left(1 + \dfrac{1}{z}\right)$ is K, then find K/10.

Consider f, g and h be three real valued functions defined on ℝ.Let f(x) = sin 3x + cos x, g(x) = cos 3x + sin x and h(x) = f²(x) + g²(x)The length of a longest interval in which the function y = h(x) is increasing, is:

A rod of length 5 has ends A and B sliding along the curve $y = 2x^2$. Let $x_A$ and $x_B$ be the x-coordinates of the ends. When A is at $(0, 0)$ and B is at $(1, 2)$, find $\frac{dx_B}{dx_A}$.

If $\log_e y=3\sin^{-1}x$, then $(1-x^2)y''-xy'$ at $x=\dfrac{1}{2}$ is equal to:

The tangent at the point $(2, -2)$ to the curve, $x^2y^2 - 2x = 4(1-y)$ does not pass through the point

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

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis, is _____.

Question 24: Given y = (x + √(1 + x²))ⁿ ... (i)Show that (1 + x²)(dy/dx)² = n²y²Hence find the value of n if d²x/dy² · (dx/dy)² is constant

In the above problem, the range of f(x)\ for x \in [-1, 1]\ is:

Curve is parametrically represented by $\begin{cases} x = \cos t + \ln\!\left(\tan\dfrac{t}{2}\right) \\ y = \sin t \end{cases}$ where $t$ is a parameter. The length of the tangent drawn to the curve at the point where its $x$-coordinates is equal to its $y$-coordinates is:

Curve is parametrically represented by $\begin{cases}x=\cos t+\ln\!\left(\tan\dfrac{t}{2}\right)\\y=\sin t\end{cases}$ where $t$ is a parameter. The length of the tangent drawn to the curve at the point where its $x$-coordinates is equal to its $y$-coordinates is:

The tangent to $y = f(x)$ at $x = 0$ has slope equal to:

Consider the triangles with vertices $A(2,1)$, $B(0,0)$ and $C(t,4)$, $t\in[0,4]$. If the maximum and the minimum perimeters of such triangles are obtained at $t=\alpha$ and $t=\beta$ respectively, then $6\alpha+21\beta$ is equal to ___________.

If x = −1 and x = 2 are extreme points of f(x) = α log |x| + βx2 + x, then

If $S_1$ and $S_2$ are respectively the sets of local minimum and local maximum points of the function, $f(x) = 9x^4 + 12x^3 - 36x^2 + 25$, $x \in \mathbb{R}$, then:

If $f(x) = x^{3} - x^{2}f'(1) + xf''(2) - f'''(3)$, $x \in \mathbb{R}$, then (1) $3f(1) + f(2) = f(3)$ (2) $f(3) - f(2) = f(1)$ (3) $2f(0) - f(1) + f(3) = f(2)$ (4) $f(1) + f(2) + f(3) = f(0)$

Let M and m be respectively the absolute maximum and the absolute minimum values of the function, $f(x) = 2x^3 - 9x^2 + 12x + 5$ in the interval [0, 3]. Then M − m is equal to

A sector of a circle of radius 1 with angle α is bent to form a cone. The volume of the vessel (cone). If α = π:

A rod of fixed length $k$ slides along the coordinate axes. If it meets the axes at $A(a, 0)$ and $B(0, b)$, then the minimum value of $\left(a + \dfrac{1}{a}\right)^2 + \left(b + \dfrac{1}{b}\right)^2$ is

Let f be a function defined on ℝ (the set of all real numbers) such that f2(x) ≤ 2010(x − 2009)(x − 2010)(x − 2011)(x − 2012) + 4 for all x ∈ ℝ. If g is a function defined on ℝ with values in the interval (0, 1) such that f(x) = ln(g(x)) for all x ∈ ℝ, then the number of points in ℝ at which g has a local maximum is ______.

Let $g(x)=3f\left(\dfrac{x}{3}\right)+f(3-x)$ and $f''(x)>0$ for all $x\in(0,3)$. If $g$ is decreasing in $(0,\alpha)$ and increasing in $(\alpha,3)$, then $8\alpha$ is:

Given $y^2 = 6x$ and $9x^2 + by^2 = 16$. If both curves intersect each other at right angles, find the value of $b$.

The angle between the curves $x^3 - 3xy^2 + 2 = 0$ and $3x^2y - y^3 - 2 = 0$ is:

The function $f(x) = \dfrac{x}{2} + \dfrac{2}{x}$ has a local minimum at

Find the minimum value of $|\sin x + \cos x + \tan x + \cot x + \sec x + \text{cosec}\, x|$, for real numbers $x$.

If $\frac{dy}{dx} = (2x^2 + 1)e^{x^2}$, find the equation of the tangent at x = 1. Which point does the tangent pass through?

Suppose $f(x)=\dfrac{(2^x+2^{-x})\tan x\sqrt{\tan^{-1}(x^2-x+1)}}{(7x^2+3x+1)^3}$. Then the value of $f'(0)$ is equal to

Applications of Derivatives Questions (514)