How to prove that $e^{ax}$ is convex for any $a \in \mathbb{R}$?
Can I prove this without using derivatives? Maybe Taylor Series?
2026-03-30 21:09:54.1774904994
How to prove that $e^{ax}$ is convex?
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By AM-GM inequality we get $$e^{a\tfrac{u+v}{2}}=\sqrt{e^{au}e^{av}}\le\frac{e^{au}+e^{av}}{2}.$$
This is Jensen inequality. Continuity concludes the proof.