1. ## 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! 2. 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?

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