Applications of Derivatives Questions - JEE

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

If non-zero real numbers $b$ and $c$ are such that $\min f(x) > \max g(x)$, where $f(x) = x^2 + 2bx + 2c^2$ and $g(x) = -x^2 - 2cx + b^2$ ($x \in \mathbb{R}$); then $\left|\dfrac{c}{b}\right|$ lies in the interval:

A spherical balloon is being inflated at a constant rate. The volume of the balloon after 49 minutes of leakage is $4500\pi - 49(72\pi) = 972\pi$. The rate (in meters per minute) at which the radius of the balloon decreases 49 min after the leakage began is:

A function $y = f(x)$ has a second order derivative $f''(x) = 6(x-1)$. If its graph passes through the point (2, 1) and at that point the tangent to the graph is $y = 3x - 5$, then the function is

$P_r(x_r, y_r);\, r = 1, 2, 3, \ldots$ is a sequence of points on the curve $y^3 = x$. The tangent at $P_n$ cuts the curve again at $P_{n+1}$ for all $n \in \mathbb{N}$. Then $y_1, y_2, y_3, \ldots$ are in

The sum of the maximum and minimum values of the function f(x) = \sin\left(\frac{1}{2x}\right) + \cos\left(\frac{1}{2x}\right) + \sec\left(\frac{1}{2x}\right) is

Let f(x) = x cos−1(−sin|x|), x ∈ [−π/2, π/2]. Which of the following is true?

Given that $x$, $y$, $z$ are positive reals such that $xyz = 32$. The minimum value of $x^2 + 4xy + 4y^2 + 2z^2$ is ______.

Given f(x) = x√(kx − x²). If f(x) is an increasing function on [0, 3] and M is the maximum value of f(x), find the minimum value of k (denoted m) such that f is increasing on [0,3], and find M. What is the value of M (approximately)?

Let $C$ be a curve given by $y(x) = 1 + \sqrt{4x - 3}$, $x > 3/4$. If $P$ is a point on $C$, such that the tangent at $P$ has slope $2/3$, then a point through which the normal at $P$ passes, is

Let f and g be two differentiable functions on R such that f'(x) > 0 and g'(x) < 0, for all x ∈ R. Then for all x:

Find the maximum and minimum values of \[ f(x) = x(1-x)^2, \quad 0 \leq x \leq 2. \]

If $f(x)$ is continuous and derivable on $[-2, 5]$ and $-4 \leq f'(x) \leq 3$ $\forall\, x \in (-2, 5)$, then difference of maximum and minimum value of $f(5)$ is equal to:

Let $ P(x) = x^4 + ax^3 + bx^2 + cx + d $ be a polynomial such that $ P'(x) = 4x^3 + 3ax^2 + 2bx + c $. Given that \( P(-1)

Find the vertical angle of a right circular cone of minimum curved surface area that circumscribes a given sphere.

Applications of Derivatives Questions (514)