# Math Help - Find the maximum and minimum values of

1. ## Find the maximum and minimum values of

Find the maximum and minimum values of on the disk D: .
maximum value:
minimum value:

2. Well, you can probably see by inspection that (x,y)=(0,0) is a minimum, but let's look at the critical points - where $\frac{\partial{f}}{\partial{x}}=\frac{\partial{f}} {\partial{y}}=0$.

$\frac{\partial{f}}{\partial{x}}=12x$

$\frac{\partial{f}}{\partial{y}}=14y$

so the only critical point is (0,0). But we have to look at the boundary, that is, where $x^2+y^2=1$. Solving for $y^2$ and substituting gives:

$6x^2+7y^2=6x^2+7(1-x^2)=7-x^2$, which has a maximum at x=0, and minima at x=1 and -1. The minima are greater than the minimum at (0,0), so we can ignore them. But the maximum at x=0 gives us maxima at (x,y)=(0,1) and (0,-1).

- Hollywood