Science:Math Exam Resources/Courses/MATH102/December 2015/Question 01/Solution 1

By the hint, all local maxima and minima on a function's graph occur at critical points of the function. (i.e., the derivative vanishes at these points.)

Since at the local local extrema of the function $h(x)$, the function $g(x)$ vanishes, we expect $h'(x)=g(x)$.

Indeed, on the interval where $h$ increases, the function values of $g$ are positive, while on the intervals where $h$ decreases, the values are negative. This is consistent with the the statement that $h'(x)=g(x)$.

In a similar way, we can find $g'(x)=f(x)$ because at the local local extrema of the function $g(x)$, the function $f(x)$ vanishes, and the intervals where $f$ has positive(negative) sign also match with the intervals where $g$ increases(decreases).

In sum, we have $f(x)=g'(x)=h''(x)$.

Answer: $\color {blue}(a)$.