Please help me out with my math homework, thank you!

Question

The figure below shows the graph of f ', the derivative of the function f, on the closed interval from x = -2 to x = 6. The graph of the derivative has horizontal tangent lines at x = 2 and x = 4.

Find the x-value where f attains its absolute minimum value on the closed interval from x = -2 to x = 6. Justify your answer. 

Answer 1

I was working on this problem also.

Maybe my version will help, too.

You need to sketch f given the information you have.

f'(x) is approximately a a cubic polynomial which means f will look like a quartic (think W)

f slopes down (negative slope) to x=2 and then slopes down again...this is a horizontal tangent.

f then slopes down to to x=5 where is has a minimum and then slopes up.

There is your answer.

Answer 2

Whenever a differentiable function is decreasing, its derivative is negative. Look at the graph of f'. It is negative all the way up until x = 5 (except at the single point x = 2). f' is positive after that, which means it is increasing for the rest of the interval.

Think about tracing the graph of the function f from left to right. Can you see the answer?