Question

A function f has f(10)=23, f’(10)=2 and f”(x)<0, for x>_10. Which of the following are possible values for f(14)?

A.35
B.31
C.27

Answer 1

f(10) = 23

f'(10) = 2, so this means at the moment x=2, the slope of f(x) = 2.

f''(x) < 0 for x ≥ 10 tells us that f'(x) is decreasing, so the slopes on f(x) are decreasing. This means that the f'(11) < 2 and f'(12) < f'(11) < 2.

If the slope of f(x) had stayed at 2 on the interval [10,14], then f(14) would equal 23 + 4(2) = 31.

Since the slope of f(x) is decreasing, we know that f(14) < 31.

The only answer that fits that is 27.