Multivariable Calculus question about the function $f(x,y)=|x|+y^2$

Question

I've got this problem in my textbook that I'm a bit unsure about. It is split in two and it reads (translated by me):

i. Explain why the function $f(x,y)=|x|+y^2$ has a greatest and smallest value on the set $D=\{(x,y)\in \rm I\!R ^2:-1\leq x\leq 1, -1\leq y\leq 1\}$.

^I have solved this problem. The next problem is the one that bugs me:

ii. Explain why the smallest value $f$ can have on the set $D$ is found on points where the gradient doesn't exist.

My only guess is that $|x|$ is not differentiable when $x=0$ and thus the gradient cannot exist. Am I on the right track?

Answer 1

The problem seems a bit vague but here's my guess on what it might mean.

You're function $f(x,y)$ is actually differentiable away from the line $x=0$. If it'd have a minimum at $f(x_0,y_0)$ such that $x_0\ne 0$ then we must have $\nabla f(x_0,y_0)=(0,0)$. Can you show why this never happens?