Mean Value Theorem

Question

Consider the function  on the interval . Find the average or mean slope of the function on this interval.  

By the Mean Value Theorem, we know there exists a  in the open interval  such that  is equal to this mean slope. For this problem, there is only one  that works. Find it.

Answer 1

Avg = (f(8)-f(2))/(8-2)

f(8) = 4√8 + 4 = 8√2 + 4

f(2)= 4√2 + 4

Avg = (8√2+4-4√2-4)/6 = (4/6)√2

f'(c) = 2/√c

2/√c = 2√2/3

Square both sides

4/c = (4/9)2

c = 9/2 = 4.5