QuestionConsider the function. (If an answer does not exist, enter DNE.) $$f(x)=\sin (x)+\sin ^{3}(x) \text { over }-\pi<x<\pi$$ (a) Determine intervals where $f$ is increasing or decreasing. (Enter your answers using interval notation.) increasing $\square$ decreasing $\square$ (b) Determine local minima and maxima of $f$. (Enter your answers as comma-separated lists.) locations of local minima $\quad x=$ $\square$ locations of local maxima $\quad x=$ $\square$

Studdy Solution
Identify local minima and maxima using critical points and the first derivative test.
- At $x = -\frac{\pi}{2}$, $f'(x)$ changes from negative to positive, indicating a local minimum. - At $x = \frac{\pi}{2}$, $f'(x)$ changes from positive to negative, indicating a local maximum.
(a) The function is increasing on the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$ and decreasing on the intervals $(-\pi, -\frac{\pi}{2})$ and $(\frac{\pi}{2}, \pi)$.
(b) The location of the local minimum is at $x = -\frac{\pi}{2}$, and the location of the local maximum is at $x = \frac{\pi}{2}$.