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:

\(\displaystyle a^{\paren {x + y} / 2}\) \(=\) \(\displaystyle \sqrt {a^{x + y} }\) Exponent Combination Laws: Power of Power: Proof 2
\(\displaystyle \) \(=\) \(\displaystyle \sqrt {a^x a^y}\) Exponent Combination Laws: Product of Powers: Proof 2
\(\displaystyle \) \(\le\) \(\displaystyle \frac {a^x + a^y} 2\) 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.

$\blacksquare$