Calculus problem regarding graph of derivative of a function

Question

This is from GRE Math test, it is clear that $f(0)<f(2)>f(4)$, but how can I determine the remaining relation?

Also, can you recommend me materials relating to these kinds of problems?

Thank you

Answer 1

HINT

• from 0 to 2, $f$ is increasing

• from 2 to 4, $f$ is decreasing

• area under the graph for $f’$ between 0 and 2 is greater

Answer 2

The derivative at 2 is zero. To the left of 2 the derivative $f'$ is positive and to the right it is negative, which means increasing to the left and decreasing to the right. So $f(2)$ is above the other values and the answer is C or E. Any standard calculus book should be a good reference (see: First Derivative Test) The Fundamental Theorem of Calculus tells us that the area from 0 to 2 is the same as $f(2)$. There's not as much "negative area" between 2 and 4, so returning to $f(4)$ doesn't bring us down as far as $f(0)$ and the answer is C.