Answer
See explanations.
Work Step by Step
a. Given $f,g$ are positive with local maxima at $x=a$ and $f’,g’$ change sign at $x=a$, we can show that $h=f\cdot g$ has a local maximum at $x=a$. Since both $f’,g’$ change signs and give maxima at $x=a$, we have $..(+)..(a)..(-)..$ for both $f’$ and $g’$.
Thus the sign change for $h’=f’g+g’f, f\gt0,g\gt0$, would be $..(+)..(a)..(-)..$, which indicates a maximum at $x=a$. Thus the answer is yes.
b. Given $f,g$ have inflection at $x=a$, we have $f’’=0,g’’=0$. However, as an inflection point is different from an extrema, it is possible that $f’\ne0,g’\ne0$.
We have:
$h’’=f’’g+f’g’+g’f’+fg’’=2f’g’\ne0$
Thus function $h(x)$ may not have an inflection at $x=a$.
