Power Function on Strictly Positive Base is Convex
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Theorem
Let $a \in \R_{>0}$ be a strictly positive real number.
Let $f: \R \to \R$ be the real function defined as:
- $\map f x = a^x$
where $a^x$ denotes $a$ to the power of $x$.
Then $f$ is convex.
Proof
Let $x, y \in \R$.
Note that, from Power of Positive Real Number is Positive: Real Number:
- $\forall t \in \R: a^t > 0$.
So:
\(\ds a^{\paren {x + y} / 2}\) | \(=\) | \(\ds \sqrt {a^{x + y} }\) | Exponent Combination Laws: Power of Power: Proof 2 | |||||||||||
\(\ds \) | \(=\) | \(\ds \sqrt {a^x a^y}\) | Exponent Combination Laws: Product of Powers: Proof 2 | |||||||||||
\(\ds \) | \(\le\) | \(\ds \frac {a^x + a^y} 2\) | Cauchy's Mean Theorem |
Hence $a^x$ is midpoint-convex.
Further, from Power Function on Strictly Positive Base is Continuous: Real Power, $a^x$ is continuous.
Thus, from Continuous Midpoint-Convex Function is Convex, $a^x$ is convex.
$\blacksquare$