1. ## Find max/min values.

Find the max and min values of f(x,y)=x^2 + y^2 + 4 over the region R={(x,y) : x^2 + 2(y^2) =< 3

2. ## Re: Find max/min values.

You want to equate the first partials to zero and solve the resulting system to find the critical points:

$\displaystyle f_x(x,y)=0$

$\displaystyle f_y(x,y)=0$

Solve the resulting system to find any critical points in the given region.

Now you want to use the second partials test for relative extrema:

Let $\displaystyle (a,b)$ be a critical point of $\displaystyle z=f(x,y)$ and suppose $\displaystyle f_{xx},\,f_{yy}$ and $\displaystyle f_{xy}$ are continuous in a rectangular region containing $\displaystyle (a,b)$.

Let $\displaystyle D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2$.

i) If $\displaystyle D(a,b)>0$ and $\displaystyle f_{xx}(a,b)>0$, then $\displaystyle f(a,b)$ is a relative minimum.

ii) If $\displaystyle D(a,b)>0$ and $\displaystyle f_{xx}(a,b)<0$, then $\displaystyle f(a,b)$ is a relative maximum.

iii) If $\displaystyle D(a,b)<0$, then $\displaystyle f(a,b)$ is not an extremum.

iv) If $\displaystyle D(a,b)=0$, then no conclusion can be drawn concerning a relative extremum.