MAT 270: Calculus 4.2

Question

Can I get step by step example of a problem that's like this one.

Suppose f(x) is continuous on [2,6] and -2 ≤ f'(x) ≤ 4 for all x in (2,6). Use the Mean Value Theorem to estimate f(6)-f(2).

*blank* ≤ f(6)-f(2) ≤ *blank*

Answer 1

By the Mean Value Theorem, there is at least one number, c, in the interval (2,6) so that

f'(c) = [f(6) - f(2)] / (6 - 2)

So, f(6) - f(2) = 4f'(c)

Since -2 ≤ f'(c) ≤ 4, -8 ≤ 4f'(c) ≤ 16

So, -8 ≤ f(6) - f(2) ≤ 16