Let $f,g: \mathbb{R}^2 \longrightarrow \mathbb{R}^2$ be defined by
\begin{align} f(x,y)&= x^4 +y^2 \\ g(x,y)&=x^4+y^2-10x^2y \end{align}
Is it possible for $f$ and $g$ to have local minima? Here I used the formula of $(rt - s^2=0)$ hence we cannot conclude whether it has points of local extremum. So what can I do to check whether it has local extrema?
Observe for $f(x, y)$, we see that the minimal value is $0$ and $(0, 0)$ gives that value. Moreover, any deviation from $(0, 0)$ will give you a positive value. Hence $(0, 0)$ is a local minimum.
In the case of $g(x, y)$, we see that \begin{align} g(x, y) = x^4+y^2-10x^2y \end{align} which means \begin{align} \nabla g(x, y) = (x(4x^2-20y), 2y-10x^2)=(0, 0) \end{align} means \begin{align} x\cdot (4x^2-20y)=&\ 0\\ 2y-10x^2=&\ 0. \end{align} Hence $(0, 0)$ is the only critical point. However, $(0, 0)$ is only a saddle since if we move in the direction $(x, -x)$ the function value is positive for small $x>0$ and in the direction $(x, x)$ the function value is negative, again for small $x>0$.