The question is as follows (apologies for the poor formatting):
Consider a function f(x): R^2 → R with the property that its Gradient and Hessian are equal to [0 0] and
[8 5 5 1]at some point xbar.
a) Find a direction ∆x such that the relation f(xbar + ε∆x) < f(xbar) holds for sufficiently small and positive values of the scalar ε.
This is a maximizing optimization problem and the objective is to find the direction. That much I know. Totally lost as to what direction to take this. A hint would help a lot!
If it is a maximization, why do you need a descent direction? Anyway, in this case as the gradient is zero, the first order does not give you a descent direction. Therefore, you need to look at the second order approximation which comes form Hessian.
If you examine the Hessian matrix you see that it has two eigenvalues, one positive and one negative. This means that this point, in which gradient has vanished, is a saddle point. So the direction of the eigenvector corresponding to negative (positive) eigenvalue is a descent (ascent) direction.