Example 5 suppose that f(0) = −7 and f '(x) ≤ 8 for all values of x. how large can f(3) possibly be? solution we are given that f is differentiable (and therefore continuous) everywhere. in particular, we can apply the mean value theorem on the interval [0, 3] . there exists a number c such that f(3) − f(0) = f '(c) 3 correct: your answer is correct. − 0 so f(3) = f(0) + 3 correct: your answer is correct. f '(c) = −7 + 3 correct: your answer is correct. f '(c). we are given that f '(x) ≤ 8 for all x, so in particular we know that f '(c) ≤ 8 correct: your answer is correct. multiplying both sides of this inequality by 3, we have 3f '(c) ≤ 24 correct: your answer is correct. , so f(3) = −7 + incorrect: your answer is incorrect. f '(c) ≤ −7 + incorrect: your answer is incorrect. = incorrect: your answer is incorrect. the largest possible value for f(3) is .