| Last time | Next time |
First of all, it really helps to plot things! (By the way, have I mentioned to you that you all can get a free copy of Mathematica? Maybe I forgot to tell you....)
Second of all, symmetry is very useful here. Symmetry is always useful. We don't talk about it enough in K-12....
For the second derivative test, if $D>0$, there are three cases (two are conclusive): $f_{xx}>0$, $f_{xx}<0$ and $f_{xx}=0$.
It seems odd to focus entirely on $f_{xx}$, but we're just making an arbitrary choice from among two: $f_{yy}$ and $f_{xx}$ must be of the same sign for $D>0$, as one can see from its definition.
$D=0$ is inconclusive, and you can see that it varies along the $y$-axis here.
I agree that plug-n-chug problems are a pain. But Archimedes solved problems with a stick in the sand, so I haven't any pity for students who have this incredible technology at their disposal.
Perhaps I've forgotten to tell you that you have the right to a free copy of Mathematica. If so, I'm very sorry. If not, I'm very sorry. But it's a very different sorry....
So use technology to make your life easy. These derivatives are painful -- so use Mathematica! We want to focus on the math, not on the ability to differentiate an ugly expression by hand. If it's that, maybe I'll issue you all sticks and we'll do all our problems in the sandbox of the Child Care facility downstairs. We'll do math the way God intended -- like Archimedes!
Let's try #48 as a Lagrange multiplier problem.