# Critical points when gradient doesn't exist

Find and classify the critical points of: $f(x,y)=3x+4y^2$

I started off by working out the partial derivatives for the gradient, which leaves me with

$\nabla f(x,y) = (3, 8y)$, and set equal to 0 so 3=0 which means the gradient for x doesn't exist, and 8y=0. This still means I have a critical point right? If I do, how do I classify it if I can't take a second derivative of it?

The gradient exists everywhere, it's just never equal to $(0, 0)$. That is, your $f$ has no critical points.