Applications of Derivatives Questions - JEE

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

The intercepts on $x$-axis made by tangents to the curve, $y = \displaystyle\int_0^x |t|\, dt,\ x \in \mathbb{R}$, which are parallel to the line $y = 2x$, are equal to

Given function is $ f(x) = \dfrac{x}{2} + \dfrac{2}{x} $. Which of the following is correct?

If the curves $y=\dfrac{1}{a}e^x$ and $y=\ln(ax)$ (where $a$ is positive) have only one point in common, then the value of $[a]$ is:[Note: $[\cdot]$ denotes the greatest integer function.]

Let $k$ and $K$ be the minimum and the maximum values of the function $f(x) = \dfrac{(1+x)^{0.6}}{1+x^{0.6}}$ in $[0, 1]$, respectively, then the ordered pair $(k, K)$ is equal to

Given the curve $y = \sqrt{x}$ and the point $\left(\dfrac{3}{2}, 0\right)$. Find the shortest distance from the point to the curve.

Given $P(x) = x^4 + ax^3 + bx^2 + cx + d$ such that $x = 0$ is the only real root of $P'(x) = 0$. If \(P(-1)

If the equation $A_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x = 0$, $a_1 \neq 0$, $n \geq 2$, has a positive root $x = \alpha$, then the equation $n a_n x^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1 = 0$ has a positive root, which is

Given Q: We have If Find dy d²x²At the top, we have the relation dy/dx = tFind the value of n such that d²x/dy² · (dx/dy)² = (constant) is given to be satisfied.

Let $f(x) = \sin x - \cos x + \ln x$. Number of roots of $f(x) = 0$ in $(0, \infty)$ is:

The slope of the tangent at the point of inflection of $y = x^3 - 3x^2 + 6x + 2009$ is equal to:

A cone of height H and base radius R is given. A cylinder of height h and radius r is inscribed in the cone. Find the value of H/h when the volume of the cylinder is maximum.

At present, a firm is manufacturing 2000 items. It is estimated that the rate of change of production P w.r.t. additional number of workers x is given by $\dfrac{dP}{dx} = 100 - 12\sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is

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

The number of common tangents to the curves $ax^2 + by^2 = 1$ and $a_1x^2 + b_1y^2 = 1$ cutting each other at right angles is given by the condition $\frac{1}{a_1} - \frac{1}{a} = \frac{1}{b_1} - \frac{1}{b}$. How many such conditions are satisfied (find the number of solutions)?

The interval in which $f(x) = 3\cos^4 x + 10\cos^3 x + 6\cos x - 3$ decreases is $x \in [0, \pi]$

Let $y=\log_e\left(\dfrac{1-x^2}{1+x^2}\right)$, $-1<x<1$. Then at $x=\dfrac{1}{2}$, the value of $225(y'-y'')$ is equal to

$f(x) = \dfrac{(x-2)(x-1)}{(x-3)}$, $\forall x > 3$. The minimum value of $f(x)$ is equal to

Find all values of α for which the curves $y = \alpha x^2 + \alpha x + \dfrac{1}{24}$ and $x = \alpha y^2 + \alpha y + \dfrac{1}{24}$ are tangent to each other.

A tangent to the curve, $y = f(x)$ at $P(x, y)$ meets $x$-axis at $A$ and $y$-axis at $B$. If $AP : BP = 1 : 3$ and $f(1) = 1$, then the curve also passes through the point

The two curves $x = a(\cos\theta + \theta\sin\theta)$ and $y = a(\sin\theta - \theta\cos\theta)$. The distance of the normal from the origin is

The rate of increase of radius of a sphere when volume is increasing at a constant rate $4\pi$ is $\dfrac{dr}{dt}\bigg|_{\text{vol}=288} = \dfrac{dr}{dt}\bigg|_{r=6}$. The rate of increase of radius when $r = 6$ is:

The absolute minimum value of the function $\frac{x + 2}{\sqrt{x^2 + 1}}$ 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 $(1, 2)$ and B is at $(0, 0)$, find $\frac{dx_B}{dx_A}$.

Let $ f'(x) = \sqrt{x} \sin x $. Then $ f'(x) = 0 $ implies $ x = 0 $ or $ \sin x = 0 $, i.e., $ x = 2\pi, \pi $. Which of the following is correct?

f(x) = max|2 sin y − x| where y ∈ ℝ. Determine the minimum value of f(x).

The possible value of |a - c|, if a2 + 2b2 is maximum, is given by

If $f(x) = f'(x)(f(f(x)))^2$, find the minimum number of roots of the equation.Given: Either $f'(x) = 0$ has at least 4 roots, $f(x) = 1$ has at least 5 roots, $f(x) = -1$ has at least 2 roots. Find total minimum number of roots.

The possible value of |a - b|, if a2 + 2b2 is maximum, is given by

If the law of linear motion of a particle is given by $s = \dfrac{1}{3}t^3 - 16t$, then the acceleration at the time when the velocity vanishes, is

Let $f(x) = \frac{\ln g(x)}{g(x)}$. Find the value of $x$ (in km/h) at which $f'(x) = 0$, given $g(x) = \left(\frac{e-1}{50}\right)x + 1$.

If the line $ax + by + c = 0$ is a normal to the curve $xy = 1$, then

The maximum value of the sum of the intercepts made by any tangent to the curve (a sin² θ, 2a sin θ) with the axes is

Find the value of c such that the line joining points A(0, 3) and B(5, −2) becomes tangent to the curve y = c/(x + 1).

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 $a$ equals

Let $P_1 = (t_1^2,\, \sqrt{a}\, t_1^3)$ and $P_2 = (t_2^2,\, \sqrt{a}\, t_2^3)$ be two points on the curve $y^2 = ax^3$. The tangent at $P_1$ passes through $P_2$. Which of the following relations holds?

Let f(x) = ax3 + bx2 + 6, where the coefficient of x and x0 are 0 and 6 respectively, f‴(x) = 24, and an extreme of f′(x) occurs at x = −1/6. Find the maximum value of f(x) for x ∈ [0, 2].

For $f(x) = 2x - \tan^{-1} x - \ln(x + \sqrt{1+x^2})$, determine the monotonicity of $f$.

Given curve $y = x^2 - 5x + 6$. The angle between the tangents drawn at the points $x = 2$ and $x = 3$ is

If $x$, $y$, and $z$ are positive real numbers and $x = \dfrac{12 - yz}{y + z}$, then maximum value of $(xyz)$ equals ______.

Two ships A and B are sailing straight away from a fixed point O along routes such that $\angle AOB$ is always 120°. At a certain instance, $OA = 8$ km, $OB = 6$ km and the ship A is sailing at the rate of 20 km/h while the ship B is sailing at the rate of 30 km/h. Then the distance between A and B is changing at the rate (in km/h)

The curve y = ex sin(π/2 − x) = ex cos x. The slope of the tangent S = ex(cos x − sin x). For what value of x in [0, 2π] is the slope of the tangent minimum?

The normal to the curve $x = a(\cos\theta + \theta\sin\theta)$, $y = a(\sin\theta - \theta\cos\theta)$ at any point '$\theta$' is such that

The tangents to the curve y = (x - 2)2 - 1 at its points of intersection with the line x - y = 3, intersect at the point

Find the number of points on the curve $y^2 = x + \sin x$ where the tangent is horizontal, given that $|y| \leq 3$.

The equation of the straight line is

The possible maximum value of a2 + 2b2 is

The number of critical points of $f(x) = \left(\int_0^x (\cos^3 t - \frac{3}{4}t^{4/3})dt\right)^2 + 4x - 2$ in $[0, 6\pi]$ is:

The function $f(x) = \frac{4}{x - 1} - \frac{9}{x + 1}$ will

If y = 3\cos(\log x) + 4\sin(\log x), then x^2 y_2 + xy_1 is equal to

Applications of Derivatives Questions (514)