1. The problem statement, all variables and given/known data http://i.minus.com/jX32eXvLm6FGu.png [Broken] 2. Relevant equations The MVT applies if 1) The function is continuous on the closed interval [a,b] such that a<b. 2) The function is differentiable on the open interval (a,b) And if the above two conditions are fulfilled then there is some point c between a and b at which the slope is equal to (f(b) - f(a)) / (b-a) 3. The attempt at a solution 1) The function is continuous for all real x. The function has a slope for all real x. 2) The function is differentiable for all x, as stated in the problem. Therefore the MVT applies. Because the MVT applies [f(7) - f(1)] / 6 = f'(c). The maximum that f'(c) can be is 5, as stated in the problem. The slope is always between 2 and 5, including the endpoints. The minimum f'(c) can be is 2. Therefore the inequality should be 12 ≤ f(7) - f(1) ≤ 30.