The Question and answer are both given below. The query I have is regarding the answer given, specifically as underlined in RED below:

1. Explain how the two critical points (-1,0) and (1,0) were obtained (underlined in RED) ?

2. Where it states "In fact, with some extra work it can be shown that these turn out to be global maxima and minima." (the LAST paragraph), please show how the global minimum and global maximum can be obtained ?

Please explain clearly showing each step as thoroughly as possible.

If you are using hand-written notes, then please ensure they are tidy and legible as untidy written notes are difficult to interpret. Alternatively use LaTeX.

Question

Find and classify the local maximum, local minimum for the function:f(x,y)=1+x2+y2x? ?

Furthermore, are any of the local optima global optima ?

Answer

So the Hessian is negative definite at (1, 0) and thus f(x, y) has a local maximum here. In fact, with some extra work it can be shown that these turn out to be global maxima and minimaThe first-order conditions are: of (1+ +y') - x(2x) 1- x2 +1/2 0 = ar (4.1) (1 + x2 + y?) (1 + x2 + 2)2 of -2ry 0 = ay (1+ 23+ 1/2)2 (4.2) These yield: r'=lty', 0=2ry. So that we have two critical points (-1, 0) and (1, 0). To classify these, we consider the Hessian: 02 f a2 f army ardy a2 f 02 f where: 87-2 + (1+ x2 +2)3' (4.3) (1+ x2+12)2 a2 f a2 f 2y 8x2y ardy ayor + (1 + x2+ 2)3' (4.4) of 2y 8.cy (1 + x2+ 12)2 + (1+x2+ 12 )3 (4.5) " At (x, y) = (-1,0), the Hessian is: D' f(-1,0) = where, for all (z, y) # (0, 0), we have: So the Hessian is positive definite at (-1,0) and thus f(x, y) has a local minimum here. " At (x, y) = (1,0), the Hessian is: D' f (1,0) = 0 where, for all (z, y) # (0,0), we have: (x y) 0 )= 2 (27 + 0')