Applications of Derivatives Questions - JEE

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.]

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)

Applications of Derivatives Questions (514)