Monotonicity

Find the intervals in which the following function  $f\left(x\right)=20-9x+6x^2-x^3$ is

$\left(a\right)$ Strictly increasing,

$\left(b\right)$ Strictly decreasing.

Find the intervals in which the function  $f\left(x\right)=2x^3-15x^2+36x+1$ is strictly decreasing. Also find the points on which the tangents are parallel to the $x$-axis.

The intervals in which the function  $f\left(x\right)=x^3-12x^2+36x+17$  would increase is:

Let  $X$ be a positive random variable. Compare $E\left[X^a\right]$ with $\left(E\right[X]{)}^a$ for all values of $a\in R$

.

The number of real solutions of the equation ${27}^{\frac{1}{x}}+{12}^{\frac{1}{x}}=2\times 8^{\frac{1}{x}}$ is

Find the curvature of $y=3x^2+3x+2$ at the point $\left(1,8\right)$.

Find the curvature of $y=4x^2+2$ at the point $\left(1,6\right)$.

Find the curvature of $y=2x^2+2x$ at the point $\left(1,4\right)$.

Find the curvature of $y=x^2+2$ at the point $\left(1,3\right)$.

Find the curvature of $y=4x^2$ at the point $\left(1,4\right)$.

If $f\left(x\right)=k{x}^3-9x^2+9x+3(k>0)$ is increasing for all $x,$ then _____

Let $f:R\to R$ be a differentiable function such that $f^{\text{'}}\left(0\right)=1$ and $f(x+y)=f\left(x\right)f\left(y\right)$ for all $x, y\in R.$

Which of the following is true?

For a real number $a$, let ${\tan }^{-1}\left(a\right)$ denote the real number $\theta ,-\frac{\pi }{2}<\theta <\frac{\pi }{2};$ such that $\tan \left(\theta \right)=a$. The function $f\left(x\right)={\tan }^{-1}\left(b{x}^2+2bx+c\right)$, where $b$ and $c$ are positive real numbers, is increasing on

The difference between the greatest and least value of $f\left(x\right)=\tan x+\cot x+\cos x$ in the interval $x\in \left[\frac{\mathrm{\pi }}{6}, \frac{\mathrm{\pi }}{4}\right]$, is

Sum of maximum & minimum values of $f\left(x\right)=x^3+2x^2+2x-1$ in interval $x\in \left[0,4\right]$ is

$f\left(x\right)=\frac{x}{{\log }_{x}e}$ is increasing on the interval $\dots \dots .;$ where $x\in R^{+}-\left\{1\right\}.$

A function $f\left(x\right)$ is defined as $f\left(x\right)=\frac{1}{4}g\left(2x^2-1\right)+\frac{1}{2}g\left(1-x^2\right)$ and $g^{\text{'}}\left(x\right)$ is increasing function, then $f\left(x\right)$ is decreasing in the interval

The interval of $x$ for which the function $g\left(x\right)=x\log x-x+1$ is positive is

Given $x\ge 0,$ such that $3^{x^4}-1+\frac{3}{3^{x^4}+1}\ge m$, then value of $m$ equals