I need to find the maximum and minimum of functions in space. I've managed to solve a few exercises, but for some of them I get stuck when trying to solve for x and y. I'll give two examples below, hoping someone can point out what I'm doing wrong.

First example:

$\displaystyle f(x,y)=3x-4x^3+12xy, where \,x\geq0, y\geq0, x+y\leq1 $

I approach this problem like this.

Find the points at the end (not sure what these are called in English?). They are

(0,0), (1,0), (0,1).

Find the maximum and minimum points inside the area. We find these by deriving the function twice; once with regards to x, once with regards to y. We then need find where they are zero.

$\displaystyle \frac{df}{dx}=3-12x^2+12y$

$\displaystyle \frac{df}{dy}=12x$

This is where I get stuck. How to proceed from here, assuming I've done right so far?

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Second example:

$\displaystyle f(x,y)=xy-ln(x^2+y^2), where \,\frac{1}{4}\leqx^2+y^2\leq4$

We see that the "extreme" points are

$\displaystyle \left(\frac{1}{2},0\right), (0, \frac{1}{2}), (2,0), (0,2)$

The two derivatives are

$\displaystyle \frac{df}{dx}=y-\frac{2x}{x^2+y^2}$

$\displaystyle \frac{df}{dy}=x-\frac{2y}{x^2+y^2}$

Again, stuck, not knowing how to proceed.

The exercies I did manage to solve made it easy for me, since adding/subtracting one to the other left me with something similiar to 2x-2y=0 etc. I know this is somewhat basic stuff, and unless I'm mistaken the exercies are created to avoid complex numbers and what not, so clearly I've missed something simple.