Answer
(a) $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$
(b) $\left(0, \frac{\pi}{4}\right) \cup\left(\frac{3 \pi}{4}, \pi\right)$
(c) $\left(0, \frac{\pi}{2} \right)$
(d) $\left(\frac{\pi}{2}, \pi\right)$
(e) $x=\frac{\pi}{2} $
Work Step by Step
\[
-\csc ^{2} x+2=f^{\prime}(x)
\]
First derivative
Zeros: $x=\frac{3 \pi}{4}(\csc x=\pm \sqrt{2})$, $x=\frac{\pi}{4}$
(a) Positive on $\left(\frac{\pi}{4}, \frac{3 \pi}{4}\right)$ so the function $f$ is increasing on this interval.
(b) Negative on $\left(\frac{3 \pi}{4}, \pi\right),$ $\left(0, \frac{\pi}{4}\right)$, so the function $f$ is decreasing on these intervals.
Properties of first derivative
\[
2 \cot x \csc ^{2} x=f^{\prime \prime}(x)
\]
Second derivative
Zeros : $x=\frac{\pi}{2}$
(c) Positive on $\left(0, \frac{\pi}{2}\right),$ so the function $f$ is concave up on this interval.
(d) Negative on $\left(\frac{\pi}{2}, \pi\right),$ so the function $f$ is concave down on this interval.
Properties of second derivative
(e) $\frac{\pi}{2}=x$
The points where the function changes from concave up to concave down or the other way around are inflection points.
