For $x \geq 0$, $y \geq 0$, prove that
$$ |x^p-y^p| \leq p|x-y|(x^{p-1}+y^{p-1}). $$
I thought it would be simple but I messed everything up. Here are my attempts.
Fixing $x > y \geq 0$, I thought about the function
$$ h(p)=x^p-y^p-p(x-y)(x^{p-1}+y^{p-1}) $$
Then $h(1)=0$ and I wanted to prove that $h(p) \leq 0$ when $p \geq 1$ by discussing its derivative but the derivative is messy offering no way out.
Also I thought about after fixing $y \geq 0$, consider the function
$$ \varphi(x)=x^p-y^p-p(x-y)(x^{p-1}+y^{p-1}) $$
where $\varphi(y)=0$ and I want to prove that $\varphi(x) \leq 0$ when $x \geq y$ yet again it's a messy way.
I also thought about restricting $p$ to $\mathbb{N}$ or $\mathbb{Q}$ using decomposition like
$$ (x^p-y^p)=(x-y)(x^{p-1}+x^{p-1}y+x^{p-2}y^2+\cdots+y^{p-1}) $$
but the number of terms does not match.
But I do believe this should be a easy question with some special background. I must have missed something critical. Any hint/solution appreciated!