Mathematics

This problem connects the derivative (rate of change) to the total change in a function using the Mean Value Theorem. Since $f'(x)$ has a lower bound, the function must increase by at least some minimum amount over the interval—and you want the smallest possible value of $f(6)$, so you should use the minimum allowed rate of change throughout. The key is linking the derivative constraint to the distance traveled: total change equals (minimum slope) × (interval width).

This problem connects the derivative (rate of change) to the total change in a function using the Mean Value Theorem. Since $f'(x)$ has a lower bound, the function must increase by at least some minimum amount over the interval—and you want the smallest possible value of $f(6)$, so you should use the minimum allowed rate of change throughout. The key is linking the derivative constraint to the distance traveled: total change equals (minimum slope) × (interval width).