Increasing and Decreasing Function

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:

Consider the following statements :

$1.$ $f\left(x\right)=\ln x$ is increasing in $\left(0, \infty \right)$

$2.$ $g\left(x\right)=e^x+e^{\frac{1}{x}}$ is decreasing in $\left(0, \infty \right)$

Which of the statements given above is/are correct?

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

Find the point on the curve $y=x^2-4x+5$ at which the tangent is parallel to the $x-$axis.

The point on the curve $y=x^2-4x+5$ at which the tangent is parallel to the $x-$axis is $?$

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,

Write the set of values of $x$ for which the function $f\left(x\right)=\sin x-x$ is increasing.

The interval in which the function $f\left(x\right)=x^2{e}^{-x}$ is increasing, is

Write the set of values of $x$ for which the function $f\left(x\right)=\sin x-x$ is increasing.

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

Find the intervals in which the function $f$ given by $f\left(x\right)=x^2-4x+6$ is decreasing.

Prove that the function given by $f\left(x\right)=\cos x$ is increasing in $\left(\pi , 2\pi \right)$.

Prove that the function given by $f\left(x\right)=\cos x$ is neither increasing nor decreasing in $\left(0, 2\pi \right)$.

Find intervals in which the function given by $f\left(x\right)=\sin 3x, x\in \left[0, \frac{\pi }{2}\right]$ is decreasing.

Find the intervals in which the function $f$ given by $f\left(x\right)=4x^3-6x^2-72x+30$ is decreasing.

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.