Given a function f(x,y)=x^3+y^3-3xy,where [x,y]^T∈R^2.

Given a function f(x,y)=x^3+y^3-3xy,where [x,y]^T∈R^2.
a) Determine the gradient vector ▽f(x,y) and the Hessian matrix ▽^2f(x,y) of f(x,y).
b) Using Taylor's Theorem,approximate f(x,y) by a quadratic function at the point [x,y]^T=[2,2]^T.
c) Show that ▽^2f(1,1) is positive definite by definition.
d) Show that ▽^2f(-2,-2) is negative definite by the principle minor test.
e) Show that ▽^2f(0,0) is indefinite by the eigenvalue test.
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