Thread: Concavity Question

Consider the function f(x) = 7x + 8x ^-1. For this function there are four important intervals: (-\infty, A], [A,B),(B,C), and [C,\infty) where A, and C are the critical numbers and the function is not defined at B.

Find A
and B
and C

For each of the following intervals, tell whether f(x) is increasing or decreasing.

(i infty, A]:
[A,B):
(B,C]:
[C,infty):


Any help will be appreciated!

By the continuity of derivatives and the Intermediate Value Theorem, we know that if is increasing and is decreasing, that is, if

and ,

and if is defined on ,then there must be a middle value such that



Therefore, between the values of at which critical points occur, can only be increasing or decreasing, but not both. As a result, we may calculate the value of at 'test points' within these regions to determine the behavior of over that region. In our case, the test points can be any points in the intervals



To find the values of , , and , we calculate . When is ? When is undefined?