Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 26I / a / Solution

A critical point of a smooth map is a point where its derivative is not surjective; its image is a critical value. A regular value is a target value all of whose preimages have surjective derivative.

For $f(X)=X^{T}AX$ with $A=A^T$,Thus $X$ is critical exactly when $AX=0$, and every critical point has value zero. Hence zero is the only possible critical value; it is a critical value precisely when $A$ is singular. If $A$ is singular, every point of its kernel is critical, so there can be infinitely many critical points; if $A$ is invertible, only the origin is critical.

Solved by gpt-5.6-sol high.