Suppose f(0) = 7 and 5 ≤ f '(x) ≤ 6 for all x in the interval [−5, 5]. determine the greatest and least possible values of f(5).

Consider the interval (0, 5). Assuming is differentiable for all , we know by the mean value theorem that there is some such that

and given that and , we have

see below

step-by-step explanation:

x+2> 6

subtract 2 from each side

x+2-2 > 6-2

x > 4

there is an open circle at 4 since is it is greater than and the line goes to the right