Q) Let $a \in R$ and let $f: R \rightarrow R$ be given by $f(x) = x^5 - 5x + a$. Then -
(A) $f(x)$ has three real roots if $a>4$
(B) $f(x)$ has only one real root if $a<-4$
(C) $f(x)$ has three real roots if $a<-4$
(D) $f(x)$ has three real roots if $-4<a<4$
In this question, I first differentiated $f(x)$ and named it another function $g(x)$.
$f'(x) = g(x) = 5x^4 - 5$
Also, $g(0) = -5$, which means it has two roots.
$\implies g(x)$ is strictly decreasing in $(-\infty, -5]$ and strictly increasing in $(5, \infty)$.
I am stuck now. I don't understand how to proceed from here. I request to whoever answering this question, to tell me how to solve this using the method I am following. I am well aware that this question can be solved in other easier ways too. However, I want to solve this using ONLY this method. Thank you.