Observe that $\frac{1}{2}\frac{1}{4},$ but $(\frac{1}{2}{)}^{\frac{1}{2}}=(\frac{1}{4}{)}^{\frac{1}{4}}.$ In this problem, we wish to show that there are infinitely many pairs of positive real numbers $(,)$ such that ${}^{}={}^{}.$

$($a$)$ Let $f(x)=x^x.$ Find and classify all critical points of $f,$ and give the intervals on which $f$ is increasing$/$ decreasing.

$($b$)$ Give a geometric argument for why the problem statement must be true.