Here are a few relevant ideas:
1.
2. If has a max or min, then does as well. Proof: assume
Then
QED.
3. The product rule for derivatives works for the dot product.
There are some hints for you.
"Show that, at a local max or min of , the vector is perpendicular to "
I think that if , then the local min/max of should be the same as , but I am not quite sure how to proceed with the rest of the question after that or how to show that is perpendicular to .