Question

How do you find decreasing interval?

Answer 1

Definition of decreasing interval:

A function $f$is said to be decreasing in the interval $I=(a,b)$if $f\left(x\right)\ge f\left(y\right)$ for every $x<y,$ and $x,y\in (a,b)$

The derivative of a given function can be used to determine if the function is increasing or decreasing on any specific intervals in its domain. If $f\text{'}\left(x\right)>0$at each point of a given interval $I$, then the function $f$can be referred to as increasing on $I$. If $f\text{'}\left(x\right)<0$at each point in an interval $I$, then the function is said to be decreasing on $I$.

For example:

Let $f\left(x\right)=x^2+4x+1$

$\Rightarrow f\text{'}\left(x\right)=2x+4$

Put $$\begin{gathered} f\text{'}\left(x\right)=0 \\ \Rightarrow 2x+4=0 \\ \Rightarrow x=-2 \end{gathered}$$

Thus intervals are $\left(-\infty ,-2\right)$ and $\left(2,\infty \right)$

Now, $-3\in \left(-\infty ,-2\right)$ and $f\text{'}(-3)=-2<0$, Thus $f$is decreasing on $\left(-\infty ,-2\right)$

and $3\in \left(2,\infty \right)$ and $f\text{'}\left(3\right)=10>0$, Thus $f$is increasing on $\left(2,\infty \right)$