Concavity Question

frozenflames · #1 $Old$ November 6th, 2009, 04:59 PM

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!

Scott H · #2 $Old$ November 6th, 2009, 05:36 PM

By the continuity of derivatives and the Intermediate Value Theorem, we know that if $f(x_1)$ is increasing and $f(x_2)$ is decreasing, that is, if

$f'(x_1) > 0$ and $f'(x_2)<0$ ,

and if $f'(x)$ is defined on $(x_1,x_2)$ ,then there must be a middle value $c$ such that

$f'(c)=0.$

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

$(-\infty,A]\;\;\;\;\;\;\;\;\;\;[A,B)\;\;\;\;\;\;\;\;\;\;(B,C]\;\;\;\;\;\;\;\;\;\;[C,\infty)$

To find the values of $A$ , $B$ , and $C$ , we calculate $f'(x)$ . When is $f'(x)=0$ ? When is $f'(x)$ undefined?