Suppose $f''(x)>0$ on $(0,4)$, $f''(x)<0$ on $(4,8)$, and $f'(4)=0$. Which statement about $f$ on $(0,8)$ is true?

Core Concept

$f''>0$ on $(0,4)$ means $f'$ increases up to $f'(4)=0$, and $f''<0$ on $(4,8)$ means $f'$ decreases from $f'(4)=0$. So $f'(x)<0$ for all $x\neq 4$ in $(0,8)$: $f$ is decreasing, and the sign change of $f''$ at $x=4$ gives a point of inflection.