show that function $f:\mathbb{R \to\mathbb{R}}$ defined by $f(x)=|x|^{p}$ for $p\geq1$ is a convex function.

Question

show that function $f:\mathbb{R \to\mathbb{R}}$ defined by $f(x)=|x|^{p}$ for $p\geq1$ is a convex function.

hint is given that use slope criterion .but how to apply slope criterion here .

please help

Answer 1

Hint: I suggest you to prove the simple following assertion

If $f: R \to I $ is convex and $g: I \to R $ nondecreasing and convex then $gof: R \to R $ is convex .

Just apply definition.