Applications of Derivatives Questions - JEE

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

Number of solution(s) of $\ln|\sin x| = -x^2$ if $x \in \left[-\frac{3\pi}{2}, \frac{3\pi}{2}\right]$ is/are:

From the function $ f $, we have $ f'(x) = \dfrac{1}{x} + 2bx + a $. It is given that $ f $ has extreme values and hence differentiable. Which of the following is a condition that must hold?

If the normal to the curve $y = f(x)$ at the point $(3, 4)$ makes an angle $\frac{3\pi}{4}$ with the positive X-axis, then $f'(3)$ is equal to

Given $y - x^{3/2} = 7$. Find the shortest distance from the point $A\left(\dfrac{1}{2}, 7\right)$ to the curve.

Paragraph for Questions 595 and 596:A quadratic polynomial $ f(x) $ with positive leading coefficient such that $ g(x) = f(\ln x) \ \forall x > 0 $. Also the curve $ y = g(x) $ satisfies the following conditions:(a) There is exactly one value for a positive number $ p $ such that $ (p, g(p)) $ is its extremum point and $ (p^2, g(p^2)) $ is its inflection point.(b) Exactly one tangent line can be drawn from the point $ (0, 0) $ to the curve $ y = g(x) $.The value of $ \dfrac{f(10)}{f(2)} $ equals:

If $y = \frac{1 + x + x^2}{1 + x + x^2}$ and $\frac{dy}{dx} = ax + b$, then

The function $f(x) = \sin^3 x - m \sin x$ is defined on open interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ and if assumes only 1 maximum value and only 1 minimum value on this interval. Then, which one of the following must be correct?

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:

The minimum number of real roots of equation P''(x))² + P'(x) · P'''(x) = 0 is

The interval in which the function $f(x)=x^x$, $x>0$, is strictly increasing is:

If the tangent at (x0, y0) to the curve x3 + y3 = a3 meets the curve again at (x1, y1), then $\frac{y_1}{y_0}$ is equal to

Find the coordinates of the point on the curve , where 0\)>, where the ordinate is minimum.

A ladder 15 m long, leans against a wall 7 m high, and a portion of the ladder protrudes over the wall such that its projection along the vertical is 3 m. How fast does the bottom start to slip away from the wall if the ladder slides down along the top edge of the wall at 2 m/s?

Let $f:(0,\infty) \to R$ be a differentiable function satisfying $f(x) + e^{f(x)} = \dfrac{2}{x} - \ln x - 1$. Find the number of integers in the range of $x$ satisfying the inequality $f(2x^2+1) - f(x^2+5) \geq f(1),\; x > 0$.

Let $f(x) = \int_0^x e^t(t - 1)(t - 2) dt$. Then $f$ strictly decreases in the interval

Let $f(x) = e^{x^2 - 4x + 3} \cdot (2x - 4)$. The minimum value of $f(x)$ on $(2, 5]$ exists at $x = 2$ and is given by $f(x) = \frac{1}{e}$. What is $f(2)$ (as a numeric value times $e$)? Find the integer answer $N$ if the minimum value is $\frac{N}{e}$.

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 height h of a right circular cone is 20 cm and is decreasing at the rate of 4 cm/s. At the same time, the radius r is 10 cm and is increasing at the rate of 2 cm/s. Find the rate of change of the volume in cm³/s ________ (up to four decimal places).

The shortest distance between the line $y = x$ and the curve $y^2 = x - 2$ is __________ (up to four decimal places).

A conical vessel is to be prepared out of a circular sheet of metal of unit radius. In order that the vessel has maximum volume, the sectorial area that must be removed from the sheet is A1 and the area of the given sheet is A2. If A2/A1 = m + √n, where m, n ∈ ℕ, then m + n is equal to.

A spherical iron ball of radius 10 cm is coated with a layer of ice of uniform thickness that melts at a rate of 50 cm³/min. When the thickness of the ice is 5 cm, then the rate at which the thickness (in cm/min) of the ice decreases, is ________ (up to four decimal places).

Ex. 17: If $D = 4(a^2 - 3b) > 0$ and $f(x_1) \cdot f(x_2) = 0$, where $x_1, x_2$ are the roots of $f'(x) = 0$, then $f(x) = x^3 + ax^2 + bx + c$

Let $(2\alpha,\alpha)$ be the largest interval in which the function $f(t)=\dfrac{|t+1|}{t^2}$, $t<0$, is strictly decreasing. Then the local maximum value of the function $g(x)=2\log_e(x-2)+\alpha x^2+4x-\alpha$, $x>2$, is _____

Let $k$ and $m$ be positive real numbers such that the function $f(x)=\begin{cases}3x^2+k\sqrt{x+1}, & 0<x<1\\mx^2+k^2, & x\geq1\end{cases}$ is differentiable for all $x>0$. Then $\dfrac{8f'(8)}{f'\!\left(\tfrac{1}{8}\right)}$ is equal to

Let $f(x)=ax^3+bx^2+cx+41$ be such that $f(1)=40$, $f'(1)=2$ and $f''(1)=4$. Then $a^2+b^2+c^2$ is equal to:

Let $f(x)=x^5+2e^{x/4}$ for all $x\in\mathbb{R}$. Consider a function $g(x)$ such that $(g\circ f)(x)=x$ for all $x\in\mathbb{R}$. Then the value of $8g'(2)$ is:

Let $f(x)=x^3+x^2f'(1)+xf''(2)+f'''(3)$, $x\in\mathbb{R}$. Then $f'(10)$ is equal to

Let $f(x)=(x+3)^2(x-2)^3$, $x\in[-4,4]$. If $M$ and $m$ are the maximum and minimum values of $f$ respectively in $[-4,4]$, then the value of $M-m$ is:

Let $f:\mathbf{R}\to\mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x)\mathrm{m}^2-2f'(x)\mathrm{m}+f''(x)=0$ in m, has two equal roots for every $x\in\mathbf{R}$. If $f(0)=1$, $f'(0)=2$, and $(\alpha,\beta)$ is the largest interval in which the function $f(\log_e x-x)$ is increasing, then $\alpha+\beta$ is equal to _____

For the function $f(x)=(\cos x)-x+1$, $x\in\mathbb{R}$, between the following two statements: (S1) $f(x)=0$ for only one value of $x$ in $[0,\pi]$. (S2) $f(x)$ is decreasing in $\left[0,\dfrac{\pi}{2}\right]$ and increasing in $\left[\dfrac{\pi}{2},\pi\right]$.

If $f(x)=\begin{cases}x^3\sin\left(\dfrac{1}{x}\right), & x\neq0\\ 0, & x=0\end{cases}$, then:

Suppose for a differentiable function $h$, $h(0)=0$, $h(1)=1$ and $h'(0)=h'(1)=2$. If $g(x)=h(e^x)e^{h(x)}$, then $g'(0)$ is equal to:

768. 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 $f(x)$.

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:

The equation of normal at any point $\phi$ to the curve $x = a\cos\phi + a\phi\sin\phi$, $y = a\sin\phi + a\phi\cos\phi$ is always at a distance of

The feasible solution for a LPP is shown in the following figure. Let $z = 3x - 4y$ be the objective function. Minimum of $z$ occurs atCorner points: $(0,0),\, (0,8),\, (4,10),\, (6,8),\, (6,5),\, (5,0)$

Applications of Derivatives Questions (514)