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 $a\in R$

.

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 $\left(1,4\right)$.

$y=x^3$ is a monotonic function. Find whether the function is increasing or decreasing.

$y=x^3$ is a monotonically increasing function in $\left[2,8\right]$. Find whether the function have an extrema at $x=4$. Find maximum value in given domain.

$y=x^2$ is a monotonically increasing function in $\left[2,8\right]$. Find whether the function have an extrema at $x=4$. Find maximum value in given domain.

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

If $y=\frac{ax-b}{\left(x-1\right)\left(x-4\right)}$ has an extremum at the point $\left(2,-1\right)$, then the values of $a$ and $b$ are

Find number of point of inflection for the following functions $f\left(x\right)=x+\sin x$ in $\left(0, 2\pi \right)$.

$2x^3-6x+5$ is an increasing function, if

The function $f\left(x\right)={\left(9-x^2\right)}^2$ increasing in

Let $f\left(x\right)=\frac{a^2-4}{a^2+2}{x}^3-3x+\sin 3$. Then for $f\left(x\right)$ to be a function with negative slope  on $\mathrm{R}$, possible values of $a$ are given by