Question #40506

This question tests the student's ability to handle complex quadratics transformed into real-valued modulus expressions.

The trap lies in the non-differentiable point (kink) introduced by $|x|$.

Students who blindly replace $|x|$ with $x$ and differentiate over all real numbers will find a false global minimum at $x = \frac{-1-\sqrt{5}}{2}$, leading to the incorrect option C.

By enforcing the domain constraint $u = |x| \ge 0$, the true global minimum is forced onto the boundary $u=0$, highlighting the danger of relying solely on $f'(x)=0$ without checking domain boundaries and points of non-differentiability.