= 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 $f(X)=X^TAX$ with $A=A^T$,
$$
Df_X(H)=2X^TAH.
$$
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.
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