Application of Derivatives

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that $f\left(\frac{x+y}{3}\right)=\frac{f(x)+f(y)}{3}$ for all $x, y \in \mathbb{R}$, and $f^{\prime}(0)=3$. Then the minimum value of the function $g(x)=3+e^x f(x...

The number of critical points of the function $f(x) = \begin{cases} |\frac{\sin x}{x}|, & x \neq 0 \\ 1, & x = 0 \end{cases}$ in the interval $(-2\pi, 2\pi)$ is equal to :

Let $f(x)$ be a polynomial of degree 5, and have extrema at $x = 1$ and $x = -1$. If $\lim\limits_{x \to 0} \left( \frac{f(x)}{x^3} \right) = -5$, then $f(2) - f(-2)$ is equal to:

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

Consider the following three statements for the function $f:(0, \infty) \rightarrow \mathbb{R}$ defined by $f(x)=\left|\log _e x\right|-|x-1|$ :
(I) $f$ is differentiable at all $x>0$.
(II) $f$ is increasing in $(0,1)$.
(III) $f$ is decreas...

Let $\alpha$ and $\beta$ respectively be the maximum and the minimum values of the function $f(\theta)=4\left(\sin ^4\left(\frac{7 \pi}{2}-\theta\right)+\sin ^4(11 \pi+\theta)\right)-2\left(\sin ^6\left(\frac{3 \pi}{2}-\theta\right)+\sin ^6...

The least value of $\left(\cos ^2 \theta-6 \sin \theta \cos \theta+3 \sin ^2 \theta+2\right)$ is

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 $f: \mathbf{R} \rightarrow \mathbf{R}$ be a twice differentiable function such that the quadratic equation $f(x) \mathrm{m}^2-2 f^{\prime}(x) \mathrm{m}+f^{\prime \prime}(x)=0$ in m , has two equal roots for every $x \in \mathbf{R}$. If...

Let $f : \mathbb{R} \rightarrow \mathbb{R}$ 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 $g(x) = f(\tan^2 x - 2 \tan x + a),\ 0 < x < \frac{\pi}{2}$....

Let the function $ f(x) = \frac{x}{3} + \frac{3}{x} + 3, x \neq 0 $ be strictly increasing in $(-\infty, \alpha_1) \cup (\alpha_2, \infty)$ and strictly decreasing in $(\alpha_3, \alpha_4) \cup (\alpha_4, \alpha_5)$. Then $ \sum\limits_{i=1...

Let $x=-1$ and $x=2$ be the critical points of the function $f(x)=x^3+a x^2+b \log _{\mathrm{e}}|x|+1, x \neq 0$.
Let $m$ and M respectively be the absolute minimum and the absolute maximum values of $f$ in the interval $\left[-2,-\frac{1}{...

Let f : ℝ \(\to\) ℝ be a polynomial function of degree four having extreme values at x = 4 and x = 5. If $ \lim\limits_{x \to 0} \frac{f(x)}{x^2} = 5 $, then f(2) is equal to :

Let $\mathrm{a}>0$. If the function $f(x)=6 x^3-45 \mathrm{a} x^2+108 \mathrm{a}^2 x+1$ attains its local maximum and minimum values at the points $x_1$ and $x_2$ respectively such that $x_1 x_2=54$, then $\mathrm{a}+x_1+x_2$ is equal to :

The shortest distance between the curves $y^2=8 x$ and $x^2+y^2+12 y+35=0$ is:

Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be a function defined by $f(x)=||x+2|-2| x \|$. If $m$ is the number of points of local minima and $n$ is the number of points of local maxima of $f$, then $m+n$ is

If the function $f(x)=2 x^3-9 a x^2+12 \mathrm{a}^2 x+1$, where $\mathrm{a}>0$, attains its local maximum and local minimum values at p and q , respectively, such that $\mathrm{p}^2=\mathrm{q}$, then $f(3)$ is equal to :

Let $\mathrm{A}(4,-2), \mathrm{B}(1,1)$ and $\mathrm{C}(9,-3)$ be the vertices of a triangle ABC . Then the maximum area of the parallelogram AFDE, formed with vertices D, E and F on the sides BC, CA and $A B$ of the triangle $A B C$ respec...

The sum of all local minimum values of the function
$$\mathrm{f}(x)=\left\{\begin{array}{lr} 1-2 x, & x<-1 \\ \frac{1}{3}(7+2|x|), & -1 \leq x \leq 2 \\ \frac{11}{18}(x-4)(x-5), & x>2 \end{array}\right.$$
is

Consider the region $R=\left\{(x, y): x \leq y \leq 9-\frac{11}{3} x^2, x \geq 0\right\}$.
The area, of the largest rectangle of sides parallel to the coordinate axes and inscribed in R , is:

Let $(2,3)$ be the largest open interval in which the function $f(x)=2 \log _{\mathrm{e}}(x-2)-x^2+a x+1$ is strictly increasing and (b, c) be the largest open interval, in which the function $\mathrm{g}(x)=(x-1)^3(x+2-\mathrm{a})^2$ is str...

If the set of all values of $a$, for which the equation $5 x^3-15 x-a=0$ has three distinct real roots, is the interval $(\alpha, \beta)$, then $\beta-2 \alpha$ is equal to _________.

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 $81 \mathrm{~cm}^3 / \mathrm{min}$ and the thickness of the ice-cream...

Let $f(x)=\int_0^{x^2} \frac{\mathrm{t}^2-8 \mathrm{t}+15}{\mathrm{e}^{\mathrm{t}}} \mathrm{dt}, x \in \mathbf{R}$. Then the numbers of local maximum and local minimum points of $f$, respectively, are :

Let the set of all positive values of \(\lambda\), for which the point of local minimum of the function \((1+x(\lambda^2-x^2))\) satisfies \(\frac{x^2+x+2}{x^2+5 x+6}<0\), be \((\alpha, \beta)\). Then \(\alpha^2+\beta^2\) is equal to ______...

Let the set of all values of \(p\), for which \(f(x)=\left(p^2-6 p+8\right)\left(\sin ^2 2 x-\cos ^2 2 x\right)+2(2-p) x+7\) does not have any critical point, be the interval \((a, b)\). Then \(16 a b\) 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 \(\left[0, \frac{\pi}{2}\right]\) and increasing in $...

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

The number of critical points of the function \(f(x)=(x-2)^{2 / 3}(2 x+1)\) is

Let \(\mathrm{A}\) be the region enclosed by the parabola \(y^2=2 x\) and the line \(x=24\). Then the maximum area of the rectangle inscribed in the region \(\mathrm{A}\) is ________.

If the function \(f(x)=2 x^3-9 \mathrm{ax}^2+12 \mathrm{a}^2 x+1, \mathrm{a}> 0\) has a local maximum at \(x=\alpha\) and a local minimum at \(x=\alpha^2\), then \(\alpha\) and \(\alpha^2\) are the roots of the equation :

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

Let \(f(x)=x^5+2 x^3+3 x+1, x \in \mathbf{R}\), and \(g(x)\) be a function such that \(g(f(x))=x\) for all \(x \in \mathbf{R}\). Then \(\frac{g(7)}{g^{\prime}(7)}\) is equal to :

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then ...

For the function
\(f(x)=\sin x+3 x-\frac{2}{\pi}\left(x^2+x\right), \text { where } x \in\left[0, \frac{\pi}{2}\right],\)
consider the following two statements :
(I) \(f\) is increasing in \(\left(0, \frac{\pi}{2}\right)\).
(II) $$f^{\prime...

Let the maximum and minimum values of \(\left(\sqrt{8 x-x^2-12}-4\right)^2+(x-7)^2, x \in \mathbf{R}\) be \(\mathrm{M}\) and \(\mathrm{m}\), respectively. Then \(\mathrm{M}^2-\mathrm{m}^2\) is equal to _________.

Let the sum of the maximum and the minimum values of the function \(f(x)=\frac{2 x^2-3 x+8}{2 x^2+3 x+8}\) be \(\frac{m}{n}\), where \(\operatorname{gcd}(\mathrm{m}, \mathrm{n})=1\). Then \(\mathrm{m}+\mathrm{n}\) is equal to :

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

$$\text { If } f(x)=\left|\begin{array}{ccc} x^3 & 2 x^2+1 & 1+3 x \\ 3 x^2+2 & 2 x & x^3+6 \\ x^3-x & 4 & x^2-2 \end{array}\right| \text { for all } x \in \mathbb{R} \text {, then } 2 f(0)+f^{\prime}(0) \text { is equal to }$$

If the function \(f:(-\infty,-1] \rightarrow(a, b]\) defined by \(f(x)=e^{x^3-3 x+1}\) is one - one and onto, then the distance of the point \(P(2 b+4, a+2)\) from the line \(x+e^{-3} y=4\) is :

Let \(f: \rightarrow \mathbb{R} \rightarrow(0, \infty)\) be strictly increasing function such that \(\lim _\limits{x \rightarrow \infty} \frac{f(7 x)}{f(x)}=1\). Then, the value of $$\lim _\limits{x \rightarrow \infty}\left[\frac{f(5 x)}{f(...

The maximum area of a triangle whose one vertex is at \((0,0)\) and the other two vertices lie on the curve \(y=-2 x^2+54\) at points \((x, y)\) and \((-x, y)\), where \(y>0\), is :

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 \(f(x)=2^x-x^2, x \in \mathbb{R}\). If \(m\) and \(n\) are respectively the number of points at which the curves \(y=f(x)\) and \(y=f^{\prime}(x)\) intersect the \(x\)-axis, then the value of \(\mathrm{m}+\mathrm{n}\) is ___________.

Consider the function \(f:\left[\frac{1}{2}, 1\right] \rightarrow \mathbb{R}\) defined by \(f(x)=4 \sqrt{2} x^3-3 \sqrt{2} x-1\). Consider the statements
(I) The curve \(y=f(x)\) intersects the \(x\)-axis exactly at one point.
(II) The curv...

The function \(f(x)=2 x+3(x)^{\frac{2}{3}}, x \in \mathbb{R}\), has

The function \(f(x)=\frac{x}{x^2-6 x-16}, x \in \mathbb{R}-\{-2,8\}\)

Let for a differentiable function $f:(0, \infty) \rightarrow \mathbf{R}, f(x)-f(y) \geqslant \log _{\mathrm{e}}\left(\frac{x}{y}\right)+x-y, \forall x, y \in(0, \infty)$. Then $\sum\limits_{n=1}^{20} f^{\prime}\left(\frac{1}{n^2}\right)$ is...

Let \(g(x)=3 f\left(\frac{x}{3}\right)+f(3-x)\) and \(f^{\prime \prime}(x)>0\) for all \(x \in(0,3)\). If \(g\) is decreasing in \((0, \alpha)\) and increasing in \((\alpha, 3)\), then \(8 \alpha\) is :

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