This was a two part question;
"Given f(x,y) = 36(y-x^2)^2+(x-1)^2, show that the hessian is positive semi-definite in the region y <= x^2 + 1/72."
I solved this I think by subbing y <= x^2 + 1/72 into f(x,y) to get;
f(x, x^2 + 1/72) <= x^2 - 2x + 145/144 (I cut out all the algebra, this was the final result)
then solving for the hessian (which was [2,0;0,0] and I think because there are no variables, its just an equality instead of inequality now), giving eigenvalues 2 and 0. This indicates positive semi-definiteness, which should indicate convexity of some form I believe. I'd appreciate some comment on this 'proof',but I do think it shows that the region is positive semi-definite. But the second part asks;
" Explain why the function is not convex in this region? " and I have no idea. To my knowledge, positive semi-definite means convexity. I would really really appreciate some form of explanation here. Thank you very much!