Question

verify that the function satisfies the hypothesis of the mean value theorem on the given interval. then find all numbers c that satisfy the conclusion of the mean value theorem.

f(x) = x/(x+2) , [1,4]

Answer 1

f(b)-f(a)/b-a = f'(c) MVT

to find f(b) and f(a), just plug endpoints into original function
f(b) = f(4) = (2/3)
f(a) = f(1) = (1/3)

(2/3)-(1/3)
----------- = f'(c)
4 - 1

(1/9) = f'(c)

next, find derivative of f(x)
f'(c) = f'(x)
product rule
(1/9) = (x)(x+2)^-1
(1/9) = (x+2)^-1 - x(x+2)^-2
(1/9) = (1/x+2) - (x/(x+2))^2
(1/9) = (1/x+2) * (1 - (x/(x+2))

(1/x+2) = (1/9) mult. ea s. by 9
(9/x+2) = 1
9 = x + 2
7 = x

I'm sure you can solve the other x