Stationary points problem

Question

I already made a first derivation of $f\left ( s,t \right )$. For $\frac{\partial f}{\partial s}=4s^{3}-2s-2t$ and for $\frac{\partial f}{\partial t}=4t^{3}-2s-2t$. I have to find the stationary points. I do know what to do. I tried put t from the first equation to the second but it does not work.

Can anyone help me?

Answer 1

We have $$4s^3-2s-2t = 0\\ 4t^3-2s-2t = 0$$ Subtracting the second from the first, we have $$s^3 = t^3$$ This gives $s = t$ (just substitute that into either equation), leading to $4 t(t^2-1) = 0$, which gives $$t= 0, 1, -1$$

Thus, we have three roots.