# What does the graph of g(x) look like given that f(x)=(d/(dx))g(x) and g(0)=0?

## Given f(x) as shown, sketch a graph of g(x) such that $f \left(x\right) = \left(\frac{d}{\mathrm{dx}}\right) g \left(x\right)$ and $g \left(0\right) = 0$

Since $g ' \left(x\right) = f \left(x\right)$, ${\int}_{0}^{x} f \left(x\right) \mathrm{dx} = g \left(x\right)$.
We should be looking for points when $f \left(x\right) = 0$ as these will be maximums and minimums. Furthermore, concavity will be determined by whether $f$ is increasing or decreasing.
$f \left(x\right) = 0$ twice. On the left one, the point will be a minimum since $f \left(x\right)$ goes from negative (decreasing) to positive (increasing). The right one will be a maximum since it goes from positive (increasing) to negative (decreasing).
There will be a point of inflection from concave up to concave down where $f$ flattens. In the end you should get a graph resembling the following.