Find the critical points of this function

Question

Let $M={(x,y,z,w)∈R^4|x^4+y^4 + z^2 + w^2 = 1}$and let $f:M \rightarrow R$ be given by$f(x,y,z,w)=x^3 - z.$

a) Show that M is a manifold.

b) Find the critical points of f.

Part a is easy but how do you do part b?

Answer 1

This is Lagrange multipliers. You want points $p\in M$ where $df_p = 0$.

Set $\tilde f(x,y,z,w) = x^3-z$ and $g(x,y,z,w) = x^4+y^4+z^2+w^2$ as functions on $\mathbb R^4$. Define $F = (\tilde f,g)\colon \mathbb R^4\to\mathbb R^2$. Since $1$ is a regular value of $g$, $p$ is a critical point of $f$ if and only if $\text{rank}(dF_p) = 1$, i.e., $d\tilde f_p = \lambda dg_p$ for some scalar $\lambda$.