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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