Find the one C that works

Question

Consider the function f(x) =1/x  on the interval [ 4 , 10 ]. Find the average or mean slope of the function on this interval. 

 

By the Mean Value Theorem, we know there exists a c in the open interval ( 4 , 10 ) such that f'( c) is equal to this mean slope. For this problem, there is only one c that works. Find it.

Answer 1

Mean slope on [4,10] = [f(10) - f(4)] / (10 - 4) =

(1/10 - 1/4) / 6 = (2/20 - 5/20) / 6 = (-3/20) / 6 = -1/40 

 

f'(x) = -1/x^2, f'(c) = -1/40  ==> 

-1/c^2 = -1/40  ==>  c^2 = 40  ==>  c = +-√40  

 

Only the positive root is in the interval [4,10] so c = 2√10