Calculus question

Question

The function g′, is shown in the graph below. Assume that g(0)=0.

What are the (x,y) coordinates of the local maxima or minima of g(x)?

( , )

I am having trouble finding y.

Answer 1

First point: We know that a local maximum (minimum) of g happens where g start decreasing after an increase (increasing after a decrease). Thus, according to this criterion, we have a local maximum at 3 and a local minimum at 6 for g.

Second point: The function g can be obtained by through the integral of g'; meaning that the value of g at b can be obtained by the area between g' and x-axis from 0 to b.

Therefore, to find the value of g(3), we need to calculate the area between the curve of g' and x-axis from 0 to 3, which is 15/2. Similarly, g(6) = 9 - 9/2 = 9/2.

So, the local maxima happens at (3, 7.5) and the local minima at (6, 4.5).