I'm not sure how to prove $f(x) = x^4$ is strictly convex using just the definition of strict convexity: $$f((1-t)x+ty) < (1-t)f(x)+tf(y)$$ for $0<t<1$. Is this just an algebra slog? If so, I can't seem to get it to work. Are there any other ways of proving a function such as this one is strictly convex?
[Math] Proving $x^4$ is strictly convex
convex optimizationconvex-analysis
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Best Answer
I assume you can prove, by definition, that $x^2$ is strictly convex and order-preserving on $\mathbb R_+$. Now, you use this result twice:
$$ \begin{aligned} ((1-t)x + ty)^4 &= (((1 - t)x + ty)^2)^2 \\ &\leq ((1-t)x^2 + ty^2)^2 &\quad (x^2 \text{ is strictly convex and order-preserving})\\ &< (1-t)(x^2)^2 + t(y^2)^2 &\quad (x^2 \text { is strictly convex}) \\ &= (1-t)x^4 + ty^4 \end{aligned} $$