Polynomials and Lipschitz function

Question

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$

I honestly cannot figure out how to start this proof. Nothing similiar is in the textbook

My attempt:

This is the defintion my book gives for a Lipschitz funciton;

Let $A \subset \Bbb{R}$ and let $f: A \rightarrow \Bbb{R}$. If there exist $K > 0$ s.t. $|f(x) - f(u)| \le K |x-u|$ $\forall x,u \in A$, then $f$ is said to be a Lipschitz function on $A$.

Let $\frac{f(M) - f(-M)}{M-(-M)} = \frac{18M}{2M} = 9$. Let $K = 10$ and we're done?

Answer 1

Hint: Given $x<y$, we have $$ f(x)-f(y)=f'(c)(x-y) $$ for some $c\in[x,y]\subset[-M,M]$. Now write down $f'(c)$ explicitly and estimate (note that $|c|\le M$).