which are impossible which are possible

Question

A function f has f(5)=23,f′(5)=1.9 and f′′(x)<0, for x≥5. Which of the following are possible values for f(5.5) and which are impossible?

(a). f(5.5)=23.95 is  ? Possible Impossible .

(b). f(5.5)=24.9 is  ? Possible Impossible .

(c). f(5.5)=23.475 is ? Possible Impossible .

Answer 1

f''(x)< 0 means it's concave down. In other words, the rate of change is decreasing: f'(x) is decreasing.

f'(5) is the rate of change at x=5. It's 1.9 .it's getting smaller. At x=5.5, the change is < 1.9/2 (.5 is 1/2 of 1), so < .85

f(5) = 23

f(5.5) < 23 + .85

f(5.5) < 23.85

a and b are impossible given the conditions shown.

c is possible.