Calculus Optimization Question

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I don't understand what is done when $S(x)=(L(x))^2$. How does that help? I thought the only way to have solved this is through finding the first derivative and using the chain rule (which they say is messy).

Thanks. Dan.

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Note that $Im(L)\subset[0,+\infty)$ and $x^2$ is increasing in $[0,+\infty)$.

In this case, $\arg\min L(x) = \arg\min L^2(x)$