Hessian matrix and analysis

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I've been trying to solve this problem for a long time, but I don't know exactly how to start:enter image description here

Translation: M is square dxd, we want to show that f belongs to C^2 in R^d. After we want to calculate the Hessian matrix and the gradient of the function f(x)

What I've been thinking is to expand M=mij and calculate by hand but I think that would take a long time.(that works well to show it belongs to C^2(R^d)). Another way I thought it was to use somehow the expansion: f(a+h)=f(a)++1/2+||h||^2*e(h) where <,>=||.||

Thank you all (sorry for bad format of equations, I'm a beginner)