Mathematics

The key is to find where $h'(x) = 0$ by recognizing that the derivative simplifies nicely. Once you compute $h'(x) = \cos x - \cos x + x\sin x = x\sin x$, you'll see that critical points occur when $x = 0, \pi, 2\pi$ (the boundaries and where $\sin x = 0$). Evaluate $h$ at each interior critical point and any boundary behavior to identify which are actual local extrema, then order them by their $y$-values rather than their $x$-coordinates.

The key is to find where $h'(x) = 0$ by recognizing that the derivative simplifies nicely. Once you compute $h'(x) = \cos x - \cos x + x\sin x = x\sin x$, you'll see that critical points occur when $x = 0, \pi, 2\pi$ (the boundaries and where $\sin x = 0$). Evaluate $h$ at each interior critical point and any boundary behavior to identify which are actual local extrema, then order them by their $y$-values rather than their $x$-coordinates.