Question

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(x)=(d/(dx))g(x)` and `g(0)=0`

Answer 1

Since `g'(x) = f(x)`, `int_0^x f(x) dx = g(x)`.

We should be looking for points when `f(x) = 0` as these will be maximums and minimums. Furthermore, concavity will be determined by whether `f` is increasing or decreasing.

`f(x) = 0` twice. On the left one, the point will be a minimum since `f(x)` 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.

Hopefully this helps!