Consider the function f(x)=2sqrtx+4 on the interval [2,8]. Find the average or mean slope of the function on this interval. ?

Question

By the Mean Value Theorem, we know there exists a c in the open interval (2,8) such that f'(c) is equal to this mean slope. For this problem, there is only one c that works. Find it. ?

Answer 1

y = 2√x + 4
mean slope is just
∆y/∆x = ((2√8+4)-(2√2+4))/(8-2) = √2/3

So, we want to find c where y'(c) = √2/3

y'(x) = 1/√x
1/√c = √2/3
c = 9/2, which is in (2,8)

If you meant f(x) = 2√(x+4) then adjust the algebra accordingly.