Find functions $f:\mathbb{R}^n\to\mathbb{R}, n\ge 2$ such that $\nabla f(x)= 0$ and $x$ is:
a) local maximizer but not global
b) saddle point
c) global minimizer
c) this is the easiest, just pick $g(x) = \frac{1}{2}x^tAx$ for a definite matrix $A$ and thus it is convex because the second derivative is $A$, and $\nabla g(0)=0$ thus $0$ is a local minimizer. Since the function is convex, it is a global one
b) I'm thinking about $(x+y)^3$ which has derivative $3(x+y)^2$ which is $0$ at $0$, but it is also $0$ at the second derivative. So it may be that $0$ is not a minimizer. I'd say that indeed it isn't because at the direction $(0,1)$ we're dealing with $(y)^3$ which has a saddle point
a) This one is a bit trickier. I know that in one dimension, if I do something with sine such that the sine oscilates between smaller and smaller numbers each time, it has minimum and maximum points that are not global. For example, $x\sin(x)$. But how do make it work in two variables?