Solution
= Solution
The <eigenvector> identity $Ab=\tfrac12b$ gives $\nabla f(-b)=-2Ab+b=0$. Since $H_f=2A$ is a <positive-definite matrix>, this is a strict <local minimum>. A global certificate follows by completing the <quadratic form>:
$$
f(x)=(x+b)^TA(x+b)-b^TAb=(x+b)^TA(x+b)-\tfrac12.
$$
<Positive definiteness> makes the first term nonnegative, with equality only at $x=-b$. Thus \b[$-b$ is the unique <global minimum>], with value $-1/2$.