Power Function is Convex Real Function
Jump to navigation
Jump to search
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.
$\blacksquare$