two inequalities, application of MVT? $$py^{p-1}(x-y) \leq x^p-y^p \leq px^{p-1}(x-y)$$
Where $0&lt;y&lt;x, \ 1\leq p &lt;\infty$

I haven't been able to prove either of these inequalities. I tried subtracting the left from the middle and trying to show the whole thing is non-negative but haven't had any success.
Hint: If $y\le x$ and $m\le \varphi&#39;(z)\le M$ for every $z\in[y,x]$, then $$(x-y)m\le \varphi(x)-\varphi(y)\le(x-y)M.$$
It is a typical application of the Mean Value Theorem to a convex function (i.e. a function whose derivative f' is an increasing function).

Mean Value Theorem: if f is continuous and its derivative exists in the interval (a,b), then

f(b)-f(a) = f'(c)(b-a)

for some c in (a,b).

If f' is an increasing function, then  a &lt; c &lt; b  implies  f'(a) &lt; f'(c) &lt; f'(b), but now (thanks to the Mean Value Theorem) you have an expression for f'(c). The last step is choosing the right convex function for your inequality. In fact, you can use the Mean Value Theorem to obtain many inequalities from many functions.