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## Homework Statement

Find the global maximum and global minimum of

[tex]f(x, y)=x^3+y^2[/tex]

in the half disk [tex]x^2+y^2\leq{1}, y\geq{0}[/tex]

This is a rather new topic for me, so I am looking for clarification for problems such as these.

## The Attempt at a Solution

First, I determined the critical points of the function.

[tex]\nabla{f(x, y)=3x^2i+2yj}[/tex] and thus has a critical point at

[tex](x, y)=(0, 0)[/tex]

Next, we will consider along the boundary of the half disk. Since the disk is a half circle, I must include the endpoints x=1, x=-1.

We have,

[tex]x=cost, y=sint, (0, \pi{)}[/tex]

We have,

[tex]f(cost, sint)=cos^3t+sin^2t=g(t)[/tex]

Moreover,

[tex]-3cos^2tsint+2sintcost=0\rightarrow{cost=2/3}[/tex]

[tex]g(cos^{-1}2/3)=0.852[/tex]

Next, checking the boundaries at t=0 and t=π

g(0)=1, g(π)=-1

Therefore, f has a global max at the boundary on the half disk of 1 at (1, 0) and a global min of -1 at (-1, 0).

However, I executed two of the tests for determining if critical points are max/mins for the point (0. 0) however all the tests were inconclusive. What else can be done here?