Problem

Given $\displaystyle f\left( x,y \right)=e^{2x^{2}+4xy^{2}-x}$,

find and classify the extremes.

Attempt

I have a problem with the second derivatives. That expression would be huge. Have I been thinking wrong? Here's my calculations:

* 2012-09-07 15.15.06.jpg

* 2012-09-07 15.15.15.jpg

* 2012-09-07 15.15.21.jpg

The method I use to classify is:

When $\displaystyle A\mbox{C}-B^{2}$ is positive, we have a min or maximum. If A>0 then it's a min, if A<0 it's a maximum. If $\displaystyle A\mbox{C}-B^{2}$ is negative, then we have a saddle point.

$\displaystyle \left(A=\frac{\partial ^{2}}{\partial x^{2}},\; B=\frac{\partial ^{2}}{\partial xy},\; \mbox{C}=\frac{\partial ^{2}}{\partial y^{2}} \right)$