How do you verify the intermediate value theorem over the interval [5/2,4], and find the c that is guaranteed by the theorem such that f(c)=6 where #f(x)=(x^2+x)/(x-1)#?

1 Answer
Oct 20, 2016

Please see the verification in the explanation section.

Explanation:

#f# is continuous on #[5/2,4]#

Proof : #f# is a rational function and rational functions are continuous on their domains. #f# is continuous at all reals except #1#. (Its domain is all reals except #1#.) Since #1# is not is #[5/2,4]#, #f# is continuous on that closed interval.

#6# is between #f(5/2)# and #f(4)

Proof: #f(5/2) = 35/6 < 36/6 = 6# and #f(4) = 20/3 > 18/3 = 6#

Therefore, there is a #c# in #(5/2,4)# with #f(c) = 6#

To find the #c# (or #c#'s), solve the equation:

#f(x) = 6#.

Discard values outside #(5/2,4)#.

Note the solutions are #2# and #3#, but only #3# is in the interval.