I have a function $f(x,y) = 2x + x^2 + y^2$ bounded by the ellipse $x^2 + 4y^2 \leq 24$
I know how to determine the extrema within the ellipse by getting the partial derivatives and setting them to zero, but I don't really understand how to determine the extrema at the boundaries of the ellipse.
I know that the parametric equation for the ellipse is $(x, y) = (a\cos(t), b\sin(t))$ where $a = \sqrt{24}$ and $b = \sqrt 6$ but I don't really know how to use this formula for determining the extrema at those coordinates.
Best Answer
The minima , in the interior of the elliptical region occurs at (-1,0) where z takes the value -1. On the boundary of the ellipse, the maxima occurs at $(2\sqrt(6,0)$ where z takes the value $4(6 + \sqrt(6)$. Notice there are three more critical points here, one is a local maxima, two are local minima .
I've updated this drawing since I discovered that I wrote down the parameters of the elliptical cylinder wrong, sorry !
Again, just for fun, assume the y coordinate is zero, then $x^2 = 24$ , so x is $2\sqrt{6}$ , yes the negative root should also be checked. And, $2x + x^2$ takes the value $24 + 4\sqrt{6}$ a local maxima.