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 $g\left(x\right)=f\left(x\right)+f\left(1-x\right)$ and $f^{"}\left(x\right)>0,x\in \left(0,1\right)$. If $g$ is decreasing in the interval $\left(0,\alpha \right)$ and increasing in the interval $\left(\alpha ,1\right)$, then ${\tan }^{-1}2\alpha +{\tan }^{-1}\left(\frac{1}{\alpha }\right)+{\tan }^{-1}\left(\frac{\alpha +1}{\alpha }\right)$ is equal to

The set of all $a\in \mathrm{ℝ}$ for which the equation $x|x-1|+|x+2|+a=0$ has exactly one real root, is

Let $f:\left[2,4\right]\to \mathrm{ℝ}$ be a differentiable function such that $\left(x{\log }_{e}x\right)f^{\text{'}}\left(x\right)+\left({\log }_{e}x\right)f\left(x\right)+f\left(x\right)\ge 1,x\in \left[2,4\right]$ with $f\left(2\right)=\frac{1}{2}$ and $f\left(4\right)=\frac{1}{2}$.
Consider the following two statements:

(A) $f\left(x\right)\le 1, \text{for all} x\in \left[2,4\right]$

(B) $f\left(x\right)\ge 1/8, \text{for all} x\in \left[2,4\right]$

Then,

The number of points, where the curve $y=x^5-20x^3+50x+2$ crosses the $x$-axis, is $?$

The function $f\left(x\right)={\tan }^{-1}\left(\sin x+\cos x\right)$ is an increasing function in

If $f\left(x\right)={\sin }^{4}x+{\cos }^{4}x+bx+c$, then find possible values of $b$ and $c$ such that $f\left(x\right)$ is monotonic.

The function $f\left(x\right)={\tan }^{-1}\left(\frac{1-x^2}{1+x^2}\right)$ is

If $f\left(x\right)=x-2\sin x$, $0\le x\le 2\mathrm{\pi }$ is increasing in the interval $\left(a\mathrm{\pi },\mathrm{b\pi }\right)$, then $a+b$ is

Which of the following graph of $f\left(x\right)$ is monotonically increasing or decreasing at the point $x=a$?

For constant number $a$, consider the function $f\left(x\right)=ax+\cos 2x+\sin x+\cos x$ on $\mathrm{ℝ}$ such that $f\left(u\right)<f\left(v\right)$ for $u<v$. If the range of $a$ for any real number $x$ is $\left(\frac{m}{n},\infty \right)$ where $m,n\in \mathrm{\mathbb{N}}$, then the minimum value of $\left(m+n\right)$ is

Interval of monotonicity of $f\left(x\right)=\frac{x}{x^2-6x-16}$ is 

Let $f\left(x\right)=x{e}^{x\left(1-x\right)}$ then $f\left(x\right)$ is:

Let $f\left(x\right)=x{\cos }^{-1}\left(-\sin \left|x\right|\right),x\in \left[-\frac{\pi }{2},\frac{\pi }{2}\right]$, then which of the following is true?

$f:R\to R$ and $g:R\to R$ be two function defined by $f\left(x\right)={\log }_e\left(x^2+1\right)-e^{-x}+1$ and $g\left(x\right)=\frac{1-2e^{2x}}{e^x}$, find the range of $\alpha$ if $f\left(g\left(\frac{{\left(\alpha -1\right)}^2}{3}\right)\right)>f\left(g\left(\alpha -\frac{5}{3}\right)\right)$

What is the interval in which the function $f\left(x\right)=\frac{x}{\cos x}$ is increasing ?

The smallest natural value of $a$ for which the function $f\left(x\right)=2^{x+1}-a\left(2^{-x}\right)$$+\left(2a+1\right)\left(\ln 2\right)x-6$ $\forall x\in R$, is increasing