Proving the Second Partials Test

mattmns

Hello. I am just wondering about how to start this problem: Prove the Second Partials Test. The book I am using gives a hint of: Compute the directional derivative of f in the direction of the unit vector u = hi + kj and complete the square.

I guess I am just having trouble even starting it. Am I supposed to define f or something of that nature to start? I just cannot see how to take the derivative of f without knowing what f is. Thanks.

edit... Well I thought about it, and I guess I could say that the directional derivative is $$f_x h + f_y k$$

But completing the square there? Do I need to take the directional derivative again? $$(f_{xx} h + f_{yx} k)h + (f_{xy} h + f_{yy} k)k$$ Then complete the square?

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