Answer
See explanation.
Work Step by Step
\[
\sec x\left(\sec ^{2} x+\tan ^{2} x\right)=f^{\prime}(x)
\]
First derivative
Zeros: None (because sec $x$ does not have real roots and the other factor only has imaginary roots)
(a) Never negative, so the function $f$ is never decreasing
(b) Positive everywhere and so the function $f$ is increasing everywhere
$f^{\prime \prime}(x)=\tan x \sec x\left(\sec ^{2} x+\tan ^{2} x\right)+\sec x(2 \tan x$
$\left.\sec ^{2} x+2 \sec x \cdot \sec x \tan x\right)$
$=\tan x \sec x\left(\tan ^{2} x+5 \sec ^{2} x\right)$
Second derivative
Zeros : $x=0$
(c) Positive on $(0, \pi / 2)$ so the function $f$ is concave up on this interval.
(d) Negative on $(-\pi / 2,0),$ so the function $f$ is concave down on this interval.
(e) $0=x$
The points where the function changes from concave up to concave down or the other way around are inflection points.
