Lipschitz condition proof

Question

Trying for find the Lipschitz constant for $f(x) = \frac{1-x^5}{7}$ on $[0,1]$

I get to:

$$|f(x)-f(y)| = \frac{1}{7}|x^5-y^5|$$

But I cannot see a way to proceed further to get to $\dots \leq K |x-y|$?

Answer 1

Without loss of generality, assume that $x>y$. By the mean value theorem, $$ \frac{x^5-y^5}{x-y}=5c^4 $$ for some $c\in(y,x)$. Hence, $$ |x^5-y^5|=5c^4|x-y|\le5|x-y|. $$