Hi Ally,
So first of all, what is the derivative telling us? It is telling us whether the function is increasing (positive derivative) or decreasing (negative derivative).
The derivative seems to be negative from x = -2 though x = 2 when it comes to 0. This means the function is decreasing over this interval. Then it is negative again between x = 2 and x = 5 so again the function is decreasing. While it is temporarily stagnant at x = 2 (slope is zero so it is momentarily flat), it is not increasing so f(-2) >= f(x) for all x between -2 and 5. Since the derivative becomes positive at the end (at x = 6), it seems the function starts increasing between 5 and 6.
Does it increase between x = 5 and x = 6 enough to "undo" all the decreasing between x = -2 and x = 5? In order to answer that, we need to use the fundamental theorem of calculus which says that the integral of a function's derivative f'(x) from x = a to x = b is just the net change in the function over that interval f(b) - f(a). The integral is just the area under the curve so the area under the graph of f'(x) in an interval will tell us how much the function increases or decreases. The integral of f'(x) between -2 and 5 is negative and so the area between the curve of f'(x) and the x-axis in that interval tells us how much the function decreases by from f(-2) to f(5). The positive area under the curve between x = 5 and x = 6 tells us how much the function increases in this interval. Although I cannot see the plot, given the points you included it is clear that the negative area between -2 and 5 is much more than the positive area between 5 and 6.
Therefore f(-2) must be the absolute maximum on the interval because f(x) only decrease between -2 and 5 and although it increases between 5 and 6, it is not enough to get back to the value f(-2).
