Question

Suppose that 2 ≤ f ' ( x ) ≤ 3 for all values of x . What are the minimum and maximum possible values of f ( 7 ) − f ( 2 ) ? ≤ f ( 7 ) − f ( 2 ) ≤

Answer 1

10 ≤ f ( 7 ) − f ( 2 ) ≤ 15

Explanation:

Integrating the given inequalities along the interval from x = 2 to x = 7 yields the minimum and maximum possible values:

`2 \leq f ' ( x )\leq 3\\\int\limits^7_2 {2} \, dx \leq \int\limits^7_2 {f'(x)} \, dx \leq\int\limits^7_2 {3} \, dx \\\\2*7-(2*2)\leq f(7)-f(2)\leq 3*7-(3*2)\\10\leq f(7)-f(2)\leq 15`

The minimum possible value is 10 and the maximum possible value is 15.