Suppose that 4 ≤ f '(x) ≤ 5 for all values of x. What are the minimum and maximum possible values of f(4) − f(1)?

Question

???  ≤ f(4) − f(1) ≤ ???

fill in the blanks

Answer 1

Using the Fundamental Theorem of Calculus, ∫_{(from a to b)} f'(x)dx = f(b) - f(a).

Since 4 ≤ f'(x) ≤ 5, ∫_{(from 1 to 4)} 4dx ≤ ∫_{(from 1 to 4)} f'(x)dx ≤ ∫_{(from 1 to 4)} 5dx

So, 12 ≤ f(4) - f(1) ≤ 15.