Applications of Derivatives Questions - JEE

The minimum value of the function $y = |x+2| + |x-2| + |x-5| + |x-8| + |x-10|$ is:

If the graph of the function $f(x) = 3x^4 + 2x^3 + ax^2 - x + 2$ is concave up for all real values of $x$, then the value of $a$ is

The least value of $\alpha \in R$ for which $4\alpha x^2 + \dfrac{1}{x} \ge 1$, for all $x > 0$, is

If the total maximum value of the function $f(x)=\left(\dfrac{\sqrt{3}e}{2\sin x}\right)^{\!\sin 2x},\ x\in\left(0,\pi\right)$, is $k$, then $k^8_e + k^8_{e^2} + k^8_{e^5}$ is equal to

Let $f(x)=4\cos^3 x+3\sqrt{3}\cos^2 x-10$. The number of points of local maxima of $f$ in the interval $(0,2\pi)$ is:

A value of $C$ for which the conclusion of Mean Value Theorem holds for the function $f(x) = \log_e x$ on the interval [1, 3] is

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

The maximum volume (in cu.m) of the right circular cone having slant height 3 m is:

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

The maximum value of $f(x) = \cos x(1 + \cos x)$ is greater than its minimum value by:

The value of c in the Lagrange's mean value theorem for the function $f(x) = x^3 - 4x^2 + 8x + 11$, when $x \in [0, 1]$ is

Let \(0

A cone is inscribed in a sphere of radius $r$. The curved surface area of the cone is maximum when $\sin\alpha = \dfrac{1}{3}$, where $\alpha$ is the semi-vertical angle. The maximum curved surface area of the cone 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:

If $5f(x)+4f\left(\dfrac{1}{x}\right)=x^2-2$, $\forall x\neq 0$ and $y=9x^2f(x)$, then $y$ is strictly increasing in:

Let f(x) and g(x) be two functions which are defined and differentiable for all x \geq x_0. If f(x_0) = g(x_0) and f'(x) > g'(x) for all x > x_0, then

We have $\dfrac{d(p(t))}{dt} = \dfrac{1}{2}p(t) - 450$. At $t = 0$, $p(0) = 50$. Find the time $t$ when $P(t) = 0$.

Let $f(x) = 3\sqrt{x-2} + \sqrt{4-x}$ be a real valued function. If $\alpha$ and $\beta$ are respectively the minimum and the maximum values of $f$, then $\alpha^2 + 2\beta^2$ is equal to:

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

Given $y = \dfrac{x}{x^2 - 3}$. The curve passes through the point $(\alpha, \beta)$ and the tangent at $(\alpha, \beta)$ is parallel to the line $2x + 6y - 11 = 0$. Then $|6\alpha + 2\beta|$ equals:

A 2 m ladder leans against a vertical wall. If the top of the ladder begins to slide down the wall at the rate 25 cm/s, then the rate (in cm/sec) at which the bottom of the ladder slides away from the wall on the horizontal ground when the top of the ladder is 1 m above the ground is ________ (up to three decimal places).

The equation of family of curves for which the length of normal at any point P is equal to the distance of P from origin, is

Ex. 14: If \(D = 4(a^2 - 3b)

101. Slope of tangent to the curve $y = 2e^x \sin\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)\cos\!\left(\dfrac{\pi}{4} - \dfrac{x}{2}\right)$ where $0 \leq x \leq 2\pi$, is minimum at $x$ is equal to:

The sum of all local minimum values of the function ⎧ 1 - 2x, x < -1 ⎪ ⎪ 1 f (x) = ⎨ (7 + 2|x|), -1 \le x \le 2 3 ⎪ ⎩ ⎪ 11 (x - 4)(x - 5), x > 2 18 is

$f(x) = \tan^{-1}(\sin x + \cos x)$ is an increasing function in

Let $f$ be a differentiable function satisfying $f(x)=1-2x+\displaystyle\int_0^x e^{(x-t)}f(t)\,\mathrm{d}t$, $x\in\mathbf{R}$ and let $g(x)=\displaystyle\int_0^x (f(t)+2)^{15}(t-4)^6(t+12)^{17}\,\mathrm{d}t$, $x\in\mathbf{R}$. If $p$ and $q$ are respectively the points of local minima and local maxima of $g$, then the value of $|p+q|$ is equal to _____

Let $f(x)=\dfrac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}$, $x\in[0,\pi]-\left\{\dfrac{\pi}{4}\right\}$, then $f\!\left(\dfrac{7\pi}{12}\right)f''\!\left(\dfrac{7\pi}{12}\right)$ is equal to

The tangent to the curve $y = xe^{x^2}$ at the point $(1, e)$, also passes through the point:

Let the tangent to the curve $x^2+2x-4y+9=0$ at the point $P(1,3)$ on it meet the $y$-axis at $A$. Let the line passing through $P$ and parallel to the line $x-3y=6$ meet the parabola $y^2=4x$ at $B$. If $B$ lies on the line $2x-3y=8$, then $(AB)^2$ is equal to _______.

A cone has slant height $l = 3$ m. The maximum volume (in cu. m) of the cone is:

When the given equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0, 1]$, then $f'(x)$ will change sign. But $f'(x) = 6x^2 + 3 > 0$, for all values of $x \in \mathbb{R}$. The number of values of $k$ for which the equation $2x^3 + 3x + k = 0$ has two distinct real roots in $[0,1]$ is:

For the parametric curve x = x(t), y = y(t), find \frac{dy}{dx} at the point corresponding to t = 2, given that at this point \frac{dy}{dt} = 4t - 2 and \frac{dx}{dt} = 2t + 3.

Let the quadratic curve passing through the point $(-1,0)$ and touching the line $y=x$ at $(1,1)$ be $y=f(x)$. Then the $x$-intercept of the normal to the curve at the point $(\alpha,\alpha+1)$ in the first quadrant is _______.

A point $ x = x_1 $ in the domain of $ f $ is said to be a stationary point if $ f'(x_1) = 0 $. Which of the following statements is/are true?(1) Every local maximum or minimum of $ f $ is a stationary point.(2) Every stationary point of $ f $ is a local maximum or minimum.(3) If $ f $ has an absolute maximum on an open interval $(a,b)$, it must occur at a stationary point.(4) $ f $ is continuous on $(a,b)$ implies $ f $ has an absolute extremum on $(a,b)$.

If the tangent to the curve, $y = x^3 + ax - b$ at the point $(1, -5)$ is perpendicular to the line, $-x + y + 4 = 0$, then which one of the following points lie on the curve?

Let $f$ be real-valued function such that $e^{-2x}f(x) = x + 3 + \displaystyle\int_0^x \dfrac{dt}{\sqrt{t^6+1}}$ for all $x \in (-1,1)$ and let $y = g(x)$ be a function whose graph is reflection of the graph of $y = f(x)$ w.r.t. line $y = x$, then $g'(3)$ is not equal to:

Consider the following three statements for the function $f:(0,\infty)\to\mathbb{R}$ defined by $f(x)=|\log_e x|-|x-1|$: (I) $f$ is differentiable at all $x>0$. (II) $f$ is increasing in $(0,1)$. (III) $f$ is decreasing in $(1,\infty)$. Then,

Let $f$ be differentiable for all $x$. If $f(1) = -2$ and $f'(x) \geq 2$ for $x \in [1, 6]$, then

Let a, b ∈ ℝ be such that the function f given by \[f(x) = \log|x| + bx^2 + ax, \quad x \neq 0\] has extreme values at x = −1 and x = 2.Statement I: f has local maximum at x = −1 and at x = 2.Statement II: $a = -\frac{1}{2}$ and $b = -\frac{1}{4}$

The height of a right circular cylinder of maximum volume inscribed in a sphere of radius 3 is:

Let $f:\mathbb{R}\to\mathbb{R}$ be a thrice differentiable function such that $f(0)=0,\ f(1)=1,\ f(2)=-1,\ f(3)=2$ and $f(4)=-2$. Then, the minimum number of zeros of $(3f'f''+ff''')(x)$ is _______.

The angle between the tangents to the curves $y = \sin x$ and $y = \cos x$ at a point of intersection is

We have $f(1) = -2$, $f'(x) \geq 2$ for all $x \in [1, 6]$. By LMVT, there exists $c \in (1, 6)$ such that $f'(c) = \frac{f(6) - f(1)}{5}$. Since $f'(x) \geq 2$ for all $x \in [1, 6]$, what is the minimum value of $f(6)$?

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:

Let a curve $y=f(x),\ x\in(0,\infty)$ pass through the points $P\!\left(1,\dfrac{3}{2}\right)$ and $Q\!\left(a,\dfrac{1}{2}\right)$. If the tangent at any point $R(b,f(b))$ to the given curve cuts the $y$-axis at the point $S(0,c)$ such that $bc=3$, then $(PQ)^2$ is equal to _____.

If the line ax + by + c = 0 is normal to the curve xy + 5 = 0, then a and b have

If f(x) − l(x) has four zeroes, where l(x) is linear and f(x) = x4 + 2x3 + cx2 + 9x + 4, then find 100b where b is the largest value of c for which the second derivative has at least two zeroes. (Hint: the second derivative f″(x) = 6x2 + 6x + c = 0 has two zeroes if and only if the discriminant 36 − 24c > 0.)

A square piece of tin of side 30 cm is to be made into a box without top by cutting a square from each corner and folding up the flaps to form a box. If the volume of the box is maximum, then its surface area (in cm$^2$) is equal to

Applications of Derivatives Questions (514)