extrema of $z = y/x$ and $z = ye^{x^2}$

Question

I am trying to find the extrema of these two functions and classify them.

a: $z = \frac{y}{x}$ b: $z = ye^{x^2}$

However, when I evaluate their gradients, I find that they are never equal to 0 or undefined on the function's domain. Is there an other way to find critical points for functions a and b? or is the answer that they have no max, min or saddle points at all?

The gradient for a is: [$-\frac{y}{x^2}, \frac{1}{x}$]

The gradient for b is: $[2xye^{x^2}, e^{x^2}]$

Answer 1

To see that it doesn't have an extrema for both function, let $x=1$, now you can let $y$ be as large as possible or as negative as you like.

And yes, you are right that they don't even have local extrema as well.

Remark: Your $a$ and $b$ are swapped.