Applications of Derivatives Questions - JEE

Let $f(x) = (e^x - a)(3ax + 1)$. Number of possible values of $a$ satisfying $f(x) \geq 0$ for all $x \in R$.

Find the equation of the straight line which is a tangent at one point and normal at another point to the curve $y = 8t^2 - 1,\, x = 4t^2 + 3$.

Let $f$ be a real valued function with $(n+1)$ derivatives at each point of $\mathbb{R}$. For each pair of real numbers $a, b$ with $a Statement-1: There is a number \(c \in (a, b)$ for which $f^{(n+1)}(c) = f(c)$Statement-2: If $h(x)$ be a derivable function such that $h(p) = h(q)$ then by Rolle's theorem $h'(d) = 0; d \in (p, q)$

Let $f(x) = \sin^3 x + \lambda \sin^2 x$, \(\frac{\pi}{2}

Let $f(x) = \dfrac{1}{\cos^2 x} + \dfrac{4}{\sin^2 x}$. The minimum value of $f(x)$ for \(0

There is a cubic polynomial $f(x)$ with values of $x$ lying in the interval $[-1, 2]$. Given the condition:(i) $f'''(x) = 24$(ii) An extreme of $f'(x)$ lies at $x = \dfrac{-1}{6}$(iii) The coefficient of $x$ and $x^0$ in $f(x)$ are 0 and 6 respectively.Find the greatest value of $f(x)$.

725. If the curve $f(x) = 3x^3 + ax^2 + bx$ where $a, b$ are non negative integers, cuts the $x$-axis at 3 distinct points. Find the minimum value of $(a+b)$.

If $y(\theta)=\dfrac{2\cos\theta+\cos2\theta}{\cos3\theta+4\cos2\theta+5\cos\theta+2}$, then at $\theta=\dfrac{\pi}{2}$, $y''+y'+y$ is equal to:

If $y = \sqrt{\frac{x}{a + \frac{x}{b + \frac{x}{a + \cdots}}}}$, then $\frac{dy}{dx}$ equals

The equation of the curve in which the perpendicular from the origin upon the tangent is equal to the abscissa of the point of contact is

From a given solid cone of height $H$, another inverted cone is carved whose height is $h$, such that its volume is maximum, then the ratio $\dfrac{H}{h}$ is equal to:

Ex. 17: Statement I Let $f(x) = x[x]$ and $[\cdot]$ denotes greatest integer function, when $x$ is not an integral, then rule for $f'(x)$ is given by $[x]$.Statement II $f'(x)$ does not exist for any $x \in \mathbb{Z}$.

In cartesian coordinates the point A is $(x_1, y_1)$, where $x_1 = 1$ on the curve $y = x^2 + x + 10$. Then the tangent at A cuts the X-axis at B. Find the value of the dot product $\vec{OA} \cdot \vec{AB}$.

Let $f(x) = x - \dfrac{1}{x+1}$ and $g(x) = x^2 - 2ax + 4$, where $a$ is a parameter. If $\forall\, x_1 \in [0,1]$ there exists some $x_2 \in [1,2]$ such that $f(x_1) \geq g(x_2)$. Then the minimum value of $a$ is:

Find the polynomial function $f(s)$ of degree 6 which satisfies $\lim_{x \to 0} \left(1 + \frac{f(x)}{x^3}\right)^{1/x} = e^2$ and has local maximum at $x = 1$ and local minima at $x = 0$ and $x = 2$.

Given that $g(x) = 2f\left(\dfrac{x^2}{2}\right) + f(6 - x^2)$, $\forall x \in R$ and $f''(x) > 0$ $\forall x \in R$, then:

Let $ f(x) = \dfrac{1}{\cos^2 x} + \dfrac{4}{\sin^2 x} $. The minimum value of $ f(x) $ for \( 0

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

Given two curves $y = 10 - x^2$ and $y = 2 + x^2$. Find the angle of intersection of the two curves at their point of intersection (2, 6).

If a right circular cone, having maximum volume, is inscribed in a sphere of radius 3 cm, then the curved surface area (in cm²) of this cone is

If $a_0, a_1, a_2, a_3$ are all positive, then $4a_0x^3 + 3a_1x^2 + 2a_2x + a_3 = 0$ has atleast one root in $(-1, 0)$, then

If the tangent to the curve $y = \dfrac{x}{x^2 - 3}$, $x \in \mathbb{R}$, $(x \neq \pm\sqrt{3})$ at a point $(\alpha, \beta) \neq (0, 0)$ on it is parallel to the line $2x + 6y - 11 = 0$, then

If $f(x) = \sec^2 x + 4\,\text{cosec}^2\,x$, then the minimum value of $f(x)$ is:

Given that $a_n x^n + a_{n-1}x^{n-1} + \cdots + a_1 x = 0$, $a_1 \neq 0$, $x \geq 2$ has a positive root $x = \alpha$, and $na_n x^{n-1} + (n-1)a_{n-1}x^{n-2} + \cdots + a_1 = 0$ has a positive root, say $\beta$. Then which of the following is true?

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:

A spherical balloon is filled with $4500\pi$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of $72\pi$ cubic meters per minute, then the rate (in meters per minute) at which the radius of the balloon decreases 49 minutes after the leakage began is

Let f(x) = x7 + 14x5 + 16x3 + 30x − 560. Find the number of real solutions.

If $f''(x)

If y = 500e^{7x} + 600e^{-7x}, then the value of \frac{d^2y}{dx^2} is

Let g:[1, 6] \to [0, \infty) be a real valued differentiable function satisfying g'(x) = \frac{2}{x + g(x)} and g(1) = 0, then the maximum value of g cannot exceed

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:

If a > b > 0 and f(C) = \frac{(a2 − b2) cos C}{a − b sin C}, then the maximum value of f(C) is

Let f(x) = \int_0^x \cos\left(\frac{t^2 + 2t + 1}{5}\right) dt, 0 \leq x \leq 2. Then f(x)

If $f(x)$ is continuous and differentiable in $[-3, 9]$ and $f'(x) \in [-2, 8]$ $\forall x \in (-3, 9)$. Let $N$ be the number of divisors of the greatest possible value of $f(9) - f(-3)$, then find the sum of digits of $N$.

If the tangent at a point $P$, with parameter $t$, on the curve $x = 4t^2 + 3$, $y = 8t^3 - 1$, $t \in \mathbb{R}$, meets the curve again at a point $Q$, then the coordinates of $Q$ are

Ex. 15: If $D = 4(a^2 - 3b) > 0$ and \(f(x_1) \cdot f(x_2)

765. If $f(x) = (x - a)(x - b)$ for $a, b \in R$, then find the minimum number of roots of equation \[\pi(f'(x))^2 \cos(\pi(f(x))) + \sin(\pi(f(x))) f''(x) = 0\] in $[\alpha, \beta]$ where $f(\alpha) = 3 = f(\beta)$ and \(\alpha

The equation of the tangents to the curve (1 + x2)y = 1 at the points of its intersection with the curve (x + 1)y = 1, is given by

A curve whose concavity is directly proportional to the logarithm of its x-coordinates at any point on the curve, is given by

Let f(x) = {1 + sin x, x ≤ 0x2 − x + 1, x > 0}. Then,

Let $f:\mathbb{R}\to\mathbb{R}$ be a twice differentiable function such that $f''(x)>0$ for all $x\in\mathbb{R}$ and $f'(a-1)=0$, where $a$ is a real number. Let $g(x)=f(\tan^2x-2\tan x+a)$, $0<x<\dfrac{\pi}{2}$. Consider the following two statements: (I) $g$ is increasing in $\left(0,\dfrac{\pi}{4}\right)$ (II) $g$ is decreasing in $\left(\dfrac{\pi}{4},\dfrac{\pi}{2}\right)$. Then,

Let $f(x)=x^{2025}-x^{2000}$, $x\in[0,1]$ and the minimum value of the function $f(x)$ in the interval $[0,1]$ be $(80)^{80}(n)^{-81}$. Then $n$ is equal to

Let $f(x) = \begin{cases} 2 - |x^2 + 5x + 6| & x \neq -2 \\ b^2 + 1 & x = -2 \end{cases}$. If $f(x)$ has a relative maximum at $x = -2$, then the complete set of values $b$ can take is:

Let $y = f(x)$ be a differentiable function on $[0, 10]$ with $f(10) = 19$ and $-4 \leq f'(x) \leq -5$ for all $x \in [0,10]$. Using LMVT, find the value of $\left\lfloor \dfrac{19 - f(0)}{10} \right\rfloor$ (or determine $f(0)$ range). What is the number of integer values in the range of $f(0)$?

If $x\sqrt{1 + y} + y\sqrt{1 + x} = 0$, then $\frac{dy}{dx}$ equals

If the surface area of a cube is increasing at a rate of 3.6 cm2/sec, retaining its shape, then the rate of change of its volume (in cm3/sec), when the length of a side of the cube is 10 cm, is:

The equation of tangent to the curve $\left(\frac{x}{a}\right)^n + \left(\frac{y}{b}\right)^n = 2$ at $(a, b)$ is

Let $f(x) = \int_{0}^{x} (t-1)(t-2)^2 \, dt$, then find a point of minimum.

The equation of tangent at M(2, 7) to the curve y = h(x), is

If $y = x(\ln x)^{\ln(\ln x)}$, then $\frac{dy}{dx}$ is equal to

Applications of Derivatives Questions (514)