Question

If f(2)=8 and f′(x) ≥ 3f′(x)≥3 for 2≤x≤7, what is the smallest possible value for f(7)?

a) 14
b) 20
c) 26
d) 32

Answer 1

The smallest possible value for f(7) is 23 or greater.

Step-by-step explanation:

To find the smallest possible value for f(7), we can use the mean value theorem. Since f(x) is differentiable on the interval [2,7], we can find a value c in that interval such that f'(c) is equal to the average rate of change of f(x) on that interval. In this case, the average rate of change is (f(7) - f(2))/(7-2) = (f(7)-8)/5. We're given that f'(x) ≥ 3, so we can say (f(7)-8)/5 ≥ 3. Solving this inequality, we get f(7) ≥ 23. Therefore, the smallest possible value for f(7) is 23 or greater.