Power Function is Convex Real Function

Theorem
Let $p \ge 1$ be a real number.

Define $f : \hointr 0 \infty \to \hointr 0 \infty$ by:


 * $\map f x = x^p$

for each $x \in \hointr 0 \infty$.

Then $f$ is a convex function.

Proof
Applying Derivative of Power twice, we have that:


 * $f$ is twice differentiable

with:


 * $\map {f''} x = p \paren {p - 1} x^{p - 2}$

for each $x \in \hointr 0 \infty$.

Since $p \ge 1$, we have:


 * $p \paren {p - 1} \ge 0$

and so:


 * $\map {f''} x \ge 0$

for each $x \in \hointr 0 \infty$.

From Real Function with Positive Derivative is Increasing:


 * $f'$ is increasing

and so from Real Function is Convex iff Derivative is Increasing:


 * $f$ is convex.