Classifying critical points based on Hessian matrix values

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In this, at the critical points, I'm getting the following values of Hessian matrix. H(0,0) = 20 H(-5/3,0) = 40/3 H(-1,2) = -16 H(-1,-2) = -16

Based on these, how did they classify the critical points as local max, local min or saddle point?

And how to check for global optimum points. By letting x and y approach infinity and then comparing the function values (which equals infinity) to function values at critical points?