Question

If f(2) = 13 and f '(x) ≥ 3 for 2 ≤ x ≤ 4, how small can f(4) possibly be?

Answer 1

The smallest f(4) can be is calculated by the minimum increase in f(x) over the interval from x = 2 to x = 4 given a slope of at least 3, resulting in f(4) being at least 19.

Step-by-step explanation:

To determine how small f(4) could be, we need to take into account the information that f(2) = 13 and f '(x) ≥ 3 for 2 ≤ x ≤ 4. The derivative f '(x) represents the slope of the function f(x), and since the slope is at least 3, the function must increase by at least 3 units for each 1 unit increase in x over the interval from 2 to 4.

The minimum increase in the value of the function as x goes from 2 to 4 can be calculated by multiplying the minimum slope by the change in x. The change in x is 4 - 2 = 2. Therefore, the minimum increase in f(x) is 3 * 2 = 6. Starting from f(2) = 13, to find the minimum possible value of f(4), we add this increase to f(2): 13 + 6 = 19.

Therefore, the smallest value that f(4) could pos given that the slope is never less than 3, is 19.

Answer 2

The slope f'(x) = [f(4) - f(2)]/(4-2)≥3,
so [f(4) - 13]/2 ≥3
f(4) -13 ≥ 6
f(4)≥19, so it can be as small as 19.