Applications of Derivatives Questions - JEE

The minimum value of the function $f(x) = x^{3/2} + x^{-3/2} - 4\left(x + \dfrac{1}{x}\right)$ for all permissible real $x$, is:

If $2a + 3b + 6c = 0$, then at least one root of the equation $ax^2 + bx + c = 0$ lies in the interval

An isosceles triangle $ ABC $ is inscribed in a circle such that $ BC $ is the base. If the area of the triangle is $ A = \dfrac{1}{2} x^2 \sin 2\theta $ where $ x $ is the equal side and $ \theta $ is the base angle, then the area is maximum when

Let f(x) = (x − 5)55 (x − 6)66. Find the number of points of local maxima, local minima, and points of inflexion, and identify the nature of critical points.Specifically, at x = 5, 6, and x = 660/121, what is the nature of each critical point?

Find the points of maxima/minima of $\int_{0}^{x^2} \frac{t^2 - 5t + 4}{2 + e^t} \, dt$.

Consider a trapezoid ABCD with AB = 8 cm perpendicular to the base, BC = 6 cm and AD = 10 cm. The maximum possible area of rectangle inscribed in the trapezoid so that one of its sides lies on the larger base of trapezoid is:

A sector of a circle of radius 1 with angle α is bent to form a cone, where $ r = 1 - \dfrac{\alpha}{2\pi} $. The value of $ r $ for which volume is maximum (when α is variable):

Locate the position and nature of any turning points of the function y = x^3 - 3x + 2.

If the sum of the base 2 logarithms of the roots of the cubic $f(x) = 0$ is 5 then the value of $'a'$ is:

942. If tangent at a point $P_1$ (other than $(0, 0)$) on the curve $y^2 = ax^3$ meets the curve again at $P_2$. The tangent at $P_2$ meets the curve again at $P_3$ and so on, then find $\displaystyle\lim_{n \to \infty} \sum_{i=1}^{n} x_i$, where $x_i$'s are abscissae of $P_i$ with $x_1 = 3$.

At the point $P(a, a^n)$ on the graph of $y = x^n$ ($n \in \mathbb{N}$) in the first quadrant a normal is drawn. The normal intersects the Y-axis at the point (0, b). If $\lim_{a \to 0} b = \frac{1}{2}$, then $n$ equals ……….

Minimum distance between the curves $y^2 = x - 1$ and $x^2 = y - 1$ is equal to:

Let $x^2 - 3x + p = 0$ have two positive roots $a$ and $b$, then minimum value of $\left(\dfrac{4}{a} + \dfrac{1}{b}\right)$ is ______.

If $y(x) = x^{x}$, $x > 0$, then $y''(2) - 2y'(2)$ is equal to (1) $8\log_{e}2 - 2$ (2) $4\log_{e}2 + 2$ (3) $4(\log_{e}2)^{2} - 2$ (4) $4(\log_{e}2)^{2} + 2$

For $f(x) = x^2 - 4|x|$ and \(g(x) = \begin{cases} \min\{f(t): -6 \leq t \leq x\}, & x \in [-6, 0] \\ \max\{f(t): 0

If Rolle's theorem holds for the function $f(x) = 2x^3 + bx^2 + bx$, $x \in [-1, 1]$, at the point $x = \dfrac{1}{2}$, then $2b + c$ equals

The minimum value of $$f(x) = \int_0^4 e^{|x-t|} \, dt$$ where $$x \in [0, 3]$$ is:

If $f: R \to R$ is a monotonic, differentiable real valued function, $a, b$ are two real numbers and $\int_{a}^{b}(f(x) + f(a))(f(x) - f(a))dx = k\int_{f(a)}^{f(b)} x(b - f^{-1}(x))dx$, then the value of $k$ is ________.

Let $y = f(x) = \sin^{3}\!\left(\frac{\pi}{3}\left(\cos\left(\frac{\pi}{3\sqrt{2}}\left(-4x^{3}+5x^{2}+1\right)^{3/2}\right)\right)\right)$. Then at $x = 1$, (1) $2y' + \sqrt{3}\pi^{2}y = 0$ (2) $2y' + 3\pi^{2}y = 0$ (3) $\sqrt{2}y' - 3\pi^{2}y = 0$ (4) $y' + 3\pi^{2}y = 0$

Let f(x) be a polynomial of degree 4 having extreme values at x = 1 and x = 2. If $\lim_{x \to 0}\left(\frac{f(x)}{x^2}+1\right)=3$, then f(−1) is equal to

For $x \ge 0$, the smallest value of the function $f(x) = \dfrac{4x^2 + 8x + 13}{6(1+x)}$ is ______.

The maximum value of the expression $\dfrac{x^m y^n}{(1+x^{2m})(1+y^{2n})}$ is:

If the volume of a spherical ball is increasing at the rate of $4\pi$ cc/s, then the rate of increase of its radius (in cm/sec), when the volume is $288\pi$ cc, is

For Problems 13–15Suppose $f(x)$ is a function satisfying the following conditions:(i) $f(0) = 2,\ f(1) = 1$,(ii) $f$ has a minimum value at $x = 5/2$,(iii) For all $x$,\[f'(x) = \begin{vmatrix} 2ax & 2ax-1 & 2ax+b+1 \\ b & b+1 & -1 \\ 2(ax+b) & 2ax+2b+1 & 2ax+b \end{vmatrix}\]Range of $f(x)$ is

If the function $\int_0^x f(t)dt - 5$ as$[x] \to 1$, where $f$ is continuous then the number of integers in the range of $p$ so that the equation $2x + \int_0^x f(t)dt = p$ has roots of opposite sign in $(-1, 1)$.

If the Rolle's theorem holds for the function $f(x) = 2x^3 + ax^2 + bx$ in the interval $[-1, 1]$ for the point $c = \dfrac{1}{2}$, then the value of $2a + b$ is

$f : \mathbb{R} \to \mathbb{R}$ **Statement 1**: $f(x) = 12x^5 - 15x^4 + 20x^3 - 30x^2 + 60x + 1$ is monotonic and surjective on $\mathbb{R}$. **Statement 2**: A continuous function defined on $\mathbb{R}$, if strictly monotonic has its range $\mathbb{R}$.

Let I be the purchase value of an equipment and V(t) be the value after it has been used for t years. The value V(t) depreciates at a rate given by differential equation $\dfrac{dV(t)}{dt} = -k(T - t)$, where $k > 0$ is a constant and T is the total life in years of the equipment. Then the scrap value V(T) of the equipment is

51. 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:

Consider the cubic $f(x) = 8x^3 + 4ax^2 + 2bx + a$ where $a, b \in \mathbb{R}$. For $a = 1$ if $y = f(x)$ is strictly increasing $\forall x \in \mathbb{R}$ then maximum range of values of $b$ is:

Let S be a square with sides of length x. If we approximate the change in size of the area of S by \frac{dA}{dx}\bigg|_{x=x_0} \cdot h, when the sides are changed from x_0 to x_0 + h, then the absolute value of the error in our approximation, is

Let $a, b \in \mathbb{R}$ be such that the function $f$ given by $f(x) = \ln|x| + bx^2 + ax,\, x \neq 0$ has extreme values at $x = -1$ and $x = 2$.Statement-1: $f$ has local maximum at $x = -1$ and at $x = 2$.Statement-2: $a = \dfrac{1}{2}$ and $b = \dfrac{-1}{4}$

If m and n are positive integers and f(x) = ∫x1 (t − a)2n(t − b)2m+1 dt, a + b, then

Consider $f(x) = \int \left(t + \frac{1}{t}\right) dt$ and $g(x) = f'(x)$ for $x \in \left[-3, -\frac{1}{2}\right]$. If P is a point on the curve $y = g(x)$ such that the tangent to this curve at P is parallel to a chord joining the points $\left(\frac{1}{2}, g\left(\frac{1}{2}\right)\right)$ and $(3, g(3))$ of the curve, then the coordinates of the point P

Let $f(x) = \begin{cases} x^3 + x^2 + 10x, & x , then at \(x = 0$, $f(x)$ is

Let \ f(x) = x^3 - 3x. The number of solutions of \ f(f(x)) = 0 \ is:

Let $y(x) = (1+x)(1+x^{2})(1+x^{4})(1+x^{8})(1+x^{16})$. Then $y' - y''$ at $x = -1$ is equal to (1) 976 (2) 464 (3) 496 (4) 944

The curves $x = a(1 + \cos\theta)$ and $y = a\sin\theta$. The equation of the normal at $\theta$ passes through the point

The total number of local maxima and local minima of the function \(f(x) = \begin{cases} (2x)^3, & 3

The function $f(x) = \int_0^x t(e^t - 1)(t - 1)(t - 2)^3(t - 3)^5 dt$ has a local maximum at x equals

If $a$, $b$, and $c$ are positive and $9a + 3b + c = 90$, then the maximum value of $(\log a + \log b + \log c)$ is (base of the logarithm is 10) ______.

For Problems 13–15Suppose $f(x)$ is a function satisfying the following conditions:(i) $f(0) = 2,\ f(1) = 1$,(ii) $f$ has a minimum value at $x = 5/2$,(iii) For all $x$,\[f'(x) = \begin{vmatrix} 2ax & 2ax-1 & 2ax+b+1 \\ b & b+1 & -1 \\ 2(ax+b) & 2ax+2b+1 & 2ax+b \end{vmatrix}\]The value of $f(2)$ is

The values of 'K' for which the point of minimum of the function $f(x) = x^2 + 1 - K$ satisfies the inequality $\frac{x^2 - x - 2}{x^2 - 5x - 6} \geq 0$, belongs to

If the functions $g(x) = x^2 + ax + b$ and $h(x) = cx - x^2$ intersect and have the same tangent line at the point $(1, 0)$, then find the value of $(b + c - a)$.

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

Ice forms on a spherical ball of radius 10 cm. If the volume of ice is increasing at the rate of 50 cm³/min, find the rate of decrease of thickness of ice when the thickness of ice is 5 cm.

If f : \mathbb{R} \to \mathbb{R}, f(x) is a differentiable bijective function, then which of the following is true?

736. Let $f(x) = (x-1)^{100}(x-2)^{2(99)}(x-3)^{3(98)}\cdots(x-100)^{100}$ where $k = \dfrac{f'(101)}{f(101)}$. Find the value of $\dfrac{k}{50} - 97$.

If $y = e^{\sin^2 x}$, then $\frac{d^2y}{dx^2}$ in terms of $x$ is

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

Applications of Derivatives Questions (514)