Question

Given that f(0)=2 and 2≤2≤f′(x)≤9 for all x in the interval [−5,5][−5,5], what are the greatest and least possible values of f(4)?

A) f(4)=11 and f(4)=6
B) f(4)=5 and f(4)=2
C) f(4)=18 and f(4)=4
D) f(4)=27 and f(4)=8

Answer 1

To find the greatest and least possible values of f(4), we consider the given information about f'(x) and the interval [-5, 5]. The maximum value occurs at x = 5 and the minimum value occurs at x = -5.

Step-by-step explanation:

To find the greatest and least possible values of f(4), we need to consider the given information about f'(x) and use it to determine the possible range of f(x).

1. Since f(0) = 2, we know that f(x) passes through the point (0, 2) on the graph.
2. The condition 2 ≤ f'(x) ≤ 9 for all x in the interval [-5, 5] tells us that the slope of f(x) on this interval is always between 2 and 9.
3. Since f(x) is continuous and increasing with a slope greater than or equal to 2 on the interval [-5, 5], we can determine that the minimum value of f(4) will occur at x = -5 (the left endpoint of the interval) and the maximum value of f(4) will occur at x = 5 (the right endpoint of the interval).

Therefore, the greatest possible value of f(4) is f(5) and the least possible value of f(4) is f(-5).