Question

You are given a function f(x) that is continuous on the closed interval [2,10] and differentiable on the open interval (2,10). If f(2)=5 and the average rate of change is bounded between -1 and 3, that is: -1≤f'(x)≤3, what is the smallest possible value for f(10)?

Answer 1

Answer: -3

Explanation:

`-1 \leq f'(x) \leq 3\\\\-\int^(10)_(2) dx \leq \int^(10)_(2) f'(x) \text{ }dx \leq 3\int^(10)_(2) dx\\\\-8 \leq f(10)-f(2) \leq 24\\\\-8 \leq f(10)-5 \leq 24\\\\-3 \leq f(10) \leq 29`