How can I know the greatest value of the function knowing its derivative?

Question

I answered this as g(5), and my justification was that at the first part of the drawing(until x = 4) $g^{'}$ is negative, so the function was decreasing. after $x=4$, $g^{'}$ is positive, so the function is increasing. but my answer was wrong, could anyone trace where is my wrong thought? I have to answer the question in no more than 2.5 minutes.

Answer 1

You are confusing the slope in the plot of $g'$ with the sign of $g'$.

Since $g'$ is positive on $[0,2]$, $g$ is increasing there. Then $g'$ is negative on $[2,5]$ so it is decreasing there. So $g(2)$ is the maximum.