Determine whether the Mean Value Theorem can be applied to f on the closed interval [a, b]. If the Mean Value Theorem can be applied, find all values c in the open interval (a, b) such that

Question

f(b)-f(a)

f'(c)= ‾‾‾‾‾‾‾‾‾‾‾

b-a

If the Mean Value Theorem cannot be applied, explain why not.

x

f(x) =‾‾‾‾‾‾ , [1, 4]

x-5

Answer 1

in the interval (1,4) such that f'(c) = [ f(4) - f(1) ] / (4 - 1) = -5/4

By the quotient rule, f'(c) = -5 / (c - 5)²

-5 / (c - 5)² = -5/4

(c - 5)² = 4

c - 5 = 2 or -2

c = 7 or 3 (Since 7 is not in the interval (1,4), 3 is the only solution).

Answer 2

f(x) = x / (x-5) is continuous on [1,4] and is differentiable on (1,4) so the Mean Value Theorem is applicable.

So, there exists at least one number, c, in the interval (1,4) such that f'(c) = [ f(4) - f(1) ] / (4 - 1) = -5/4

By the quotient rule, f'(c) = -5 / (c - 5)²

So, -5 / (c - 5)² = -5/4

(c - 5)² = 4

c - 5 = 2 or -2

c = 7 or 3 (Since 7 is not in the interval (1,4), 3 is the only solution).