Positive Power Function on Non-negative Reals is Strictly Increasing

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Theorem

Let $a \in \Q_{> 0}$ be a strictly positive rational number.

Let $f_a: \R_{\ge 0} \to \R$ be the real function defined as:

$\map {f_a} x = x^a$

Then $f_a$ is strictly increasing.

Real Index

If $a \in \R_{>0}$ is a strictly positive real number, then the same result applies. Just use the real index variations of the theorems used to prove this one.

However, this result is specifically stated for a rational index, as this page is used to prove something else.

Proof

By the power rule for derivatives:

$\map {D_x} {x^a} = a x^{a - 1}$

By power of positive real number is positive, it is seen that:

$x > 0 \implies x^{a - 1} > 0$

By Strictly Positive Real Numbers are Closed under Multiplication, it follows that $\map {D_x} {x^a} > 0$ for all $x \in \openint 0 {+\infty}$.

Hence by Derivative of Monotone Function, $f_a$ is strictly increasing

$\blacksquare$