Intervals on which function is increasing and decreasing

Let $p(x)={x}^{5}-{q}^{2}x-q$, where q is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below.

We compute ${p}^{\mathrm{\prime}}(x)=5{x}^{4}-{q}^{2}$ and look for the critical points.

$5{x}^{4}-{q}^{2}=0\u27fax=\pm \frac{\sqrt{q}}{\sqrt[4]{5}}$

Hence we have to investigate the behavior of p′(x) for each of these intervals $(-\mathrm{\infty},-\frac{\sqrt{q}}{\sqrt[4]{5}})$, $(-\frac{\sqrt{q}}{\sqrt[4]{5}},\frac{\sqrt{q}}{\sqrt[4]{5}})$ and $(\frac{\sqrt{q}}{\sqrt[4]{5}},\mathrm{\infty})$ this will indicate when the function will be increasing and decreasing. How can this be determined when the expression $\frac{\sqrt{q}}{\sqrt[4]{5}}$ contains a prime number???

The answer should be : the function will be increasing for $x<\frac{\sqrt{q}}{\sqrt[4]{5}}$ and strictly decreasing for $-\frac{\sqrt{q}}{\sqrt[4]{5}}<x<\frac{\sqrt{q}}{\sqrt[4]{5}}$ and strictly increasing again for $x>\frac{\sqrt{q}}{\sqrt[4]{5}}$.

Let $p(x)={x}^{5}-{q}^{2}x-q$, where q is a prime number. I want to understand how to determine when the function will be decreasing and increasing on the intervals given below.

We compute ${p}^{\mathrm{\prime}}(x)=5{x}^{4}-{q}^{2}$ and look for the critical points.

$5{x}^{4}-{q}^{2}=0\u27fax=\pm \frac{\sqrt{q}}{\sqrt[4]{5}}$

Hence we have to investigate the behavior of p′(x) for each of these intervals $(-\mathrm{\infty},-\frac{\sqrt{q}}{\sqrt[4]{5}})$, $(-\frac{\sqrt{q}}{\sqrt[4]{5}},\frac{\sqrt{q}}{\sqrt[4]{5}})$ and $(\frac{\sqrt{q}}{\sqrt[4]{5}},\mathrm{\infty})$ this will indicate when the function will be increasing and decreasing. How can this be determined when the expression $\frac{\sqrt{q}}{\sqrt[4]{5}}$ contains a prime number???

The answer should be : the function will be increasing for $x<\frac{\sqrt{q}}{\sqrt[4]{5}}$ and strictly decreasing for $-\frac{\sqrt{q}}{\sqrt[4]{5}}<x<\frac{\sqrt{q}}{\sqrt[4]{5}}$ and strictly increasing again for $x>\frac{\sqrt{q}}{\sqrt[4]{5}}$.