Question

Assume f is a continuous function which is differentiable on the interval (1, 9). If f(9) = 0 and f′(x) ≥ 8 for all x, what is the largest possible value of f(1)? Justify your solution.

Answer 1

The largest possible value of f(1) is 0.

Step-by-step explanation:

Since f(x) is a continuous function that is differentiable on the interval (1, 9) and f'(x) is greater than or equal to 8 for all x, we can conclude that the function is strictly increasing in that interval. Since f(9) = 0, this means that f(x) is negative for x less than 9, and f(x) is positive for x greater than 9.

To find the largest possible value of f(1), we need to find the largest possible value of f(x) for x less than 9. Since f(x) is strictly increasing in the interval (1, 9), the largest possible value of f(x) for x less than 9 is f(9) = 0.

Therefore, the largest possible value of f(1) is 0.