Suppose f:R^N -> R is twice differentiable. Prove that f is convex if and only if its Hessian gradiant^2 f(x) is nonnegative.(adsbygoogle = window.adsbygoogle || []).push({});

How do I go about proving this? and my professor said I only need to consider when N=1. so R->R.

any help would be greatly appreciated.

For proving it backwards, this is what i have, but i am not sure if it is correct.

If the Hessian of F is nonnegative definite, then the function is locally strictly convex. A function that is locally strictly convex everywhere is strictly convex.

and I am not sure how to prove it other way

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Variational methods - prove f is convex in R->R

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**