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

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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.

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
 

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