Given the function #f(x)=(x+1)/x#, how do you determine whether f satisfies the hypotheses of the Mean Value Theorem on the interval [1/2, 2] and find the c?

1 Answer
Jim H
Nov 11, 2017

Please see below.

Explanation:

Hypothesis : #f# is continuous on #[1/2,2]#

True. #f# is a rational function, so #f# is continuous on its domain. The domain is all reals except #0# and #0# is not in #[1/2,2]#.

Hypothesis: #f# is differentiable on #(1/2,2)#

True. Remember that "differentiable" means "the derivative exists".
We have #f'(x)=-1/x^2#, so #f'(x)# exists for all reals except #0# and #0# is not in #(1/2,2)#.

The conclusion says that there is a #c# in #(1/2,2)# with

#f'(c) = (f(2)-f(1/2))/(2-(1/2))#.

To find those #c#'s, solve the equation. Discard solutions outside of #(1/2,2)#.

#-1/x^2 = -1# has solutions #x = +-1#.

So #c = 1#

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