
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! (Talking)

By the continuity of derivatives and the Intermediate Value Theorem, we know that if $\displaystyle f(x_1)$ is increasing and $\displaystyle f(x_2)$ is decreasing, that is, if
$\displaystyle f'(x_1) > 0$ and $\displaystyle f'(x_2)<0$,
and if $\displaystyle f'(x)$ is defined on $\displaystyle (x_1,x_2)$,then there must be a middle value $\displaystyle c$ such that
$\displaystyle f'(c)=0.$
Therefore, between the values of $\displaystyle x$ at which critical points occur, $\displaystyle f(x)$ can only be increasing or decreasing, but not both. As a result, we may calculate the value of $\displaystyle f'(x)$ at 'test points' within these regions to determine the behavior of $\displaystyle f(x)$ over that region. In our case, the test points can be any points in the intervals
$\displaystyle (\infty,A]\;\;\;\;\;\;\;\;\;\;[A,B)\;\;\;\;\;\;\;\;\;\;(B,C]\;\;\;\;\;\;\;\;\;\;[C,\infty)$
To find the values of $\displaystyle A$, $\displaystyle B$, and $\displaystyle C$, we calculate $\displaystyle f'(x)$. When is $\displaystyle f'(x)=0$? When is $\displaystyle f'(x)$ undefined?