Increasing and Decreasing Functions

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

Find the values of $x$ for which $y=\left[x\right(x-2){]}^2$ is an increasing function.

Find the intervals in which the following function is strictly increasing or decreasing:
$(x+1{)}^3(x-3{)}^3$

Find the intervals in which the following function is strictly increasing or decreasing:
$6-9x-x^2$

Find the intervals in which the following function is strictly increasing or decreasing:
$-2x^3-9x^2-12x+1$

Find the intervals in which the following function is strictly increasing or decreasing:
$10-6x-2x^2$

Find the intervals in which the following function is strictly increasing or strictly decreasing:
$x^2+2x-5$

Find the intervals in which the function $f$ given by $f\left(x\right)=2x^3-3x^2-36x+7$ is strictly increasing.

Find the intervals in which the function $f$ given by $f\left(x\right)=x^3+\frac{1}{x^3},x\ne 0$ is decreasing.

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

Show that the function $f$ given by $f\left(x\right)=\sin x$ is strictly decreasing in $\left(\frac{\pi }{2},\pi \right)$.

Show that the function $f$ given by $f\left(x\right)=\sin x$ is strictly increasing in $\left(0,\frac{\pi }{2}\right).$

Find the intervals in which the function $f$ given by $f\left(x\right)=\frac{4\mathrm{s}\mathrm{i}\mathrm{n}x-2x-x\mathrm{c}\mathrm{o}\mathrm{s}x}{2+\mathrm{c}\mathrm{o}\mathrm{s}x}$ is decreasing.

Which of the following function are strictly decreasing on $\left(0, \frac{\pi }{2}\right)?$

The interval in which $y=x^2{e}^{-x}$ is strictly increasing is

Prove that the function given by $f\left(x\right)=x^3-3x^2+3x-100$ is increasing in $\mathrm{R}.$

Prove that the function $f$ given by $f\left(x\right)=\log \left(\cos x\right)$ is strictly decreasing on $\left(0,\frac{\pi }{2}\right)$ and strictly increasing on $\left(\frac{\pi }{2},\pi \right)$.

Prove that the function $f$ given by $f\left(x\right)=\log \sin x$ is strictly increasing on $\left(0,\frac{\pi }{2}\right)$ and strictly decreasing on $\left(\frac{\pi }{2},\pi \right)$.

Let $I$ be any interval disjoint from $[-1,1].$ Prove that the function $f$ given by $f\left(x\right)=x+\frac{1}{x}$ is strictly increasing on $I.$

Find the least value of $a$ such that the function $f$ given by $f\left(x\right)=x^2+ax+1$ is strictly increasing on $(1,2).$