Convexity and stationary points

Problem

Compare $f(w)=w^2+2w+5$ and $g(w)=w^4-w^2$ . Identify stationary points and state which objective is convex on the real line.

Reveal answer or reference solution

$f$ has stationary point $w=-1$ and is strictly convex because $f''(w)=2$ . The stationary points of $g$ are $w=0$ and $w=\pm\frac{1}{\sqrt{2}}$ ; $g$ is not globally convex because $g''(w)=12w^2-2$ is negative near zero.

Local history

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Optimization Practice