# Convex smooth function with a single critical point must be global minimum?

Let $f\colon \mathbb{R}^n \to \mathbb{R}$ be a convex smooth function such that $f$ has a unique critical point. Is that point necessarily a global minimum? If it isn't, which other hypothesis should I add to the current ones to get this result?

• Please consult this Wiki - Second partial derivative test. – bubububub Jul 3 '18 at 12:14
• Smoothness and convexity is sufficient. Any critical point (even if multiple points exist) will be a global minimum. – Theo Bendit Jul 3 '18 at 12:32
• Smoothness is not necessary; a critical point of a convex function is globally optimal. – LinAlg Jul 3 '18 at 12:56
• Convexity and smoothness are sufficient to conclude this. You may even be able to relax the smoothness/critical point criteria a little and still get the result. – CyclotomicField Jul 3 '18 at 12:56