Question

Let f(x)=x−8x+8
f
(
x
)
=
x
−
8
x
+
8
. Find the open intervals on which f
f
is increasing (decreasing). Then determine the x
x
-coordinates of all relative maxima (minima).

Answer 1

For finding the intervals where function is increasing or decreasing

we first differentiate the function and we get f'(x),

and for increasing interval we solve the inequality f'(x)>0,

and for decreasing interval we solve the inequality f'(x)<0

and for finding the maxima or minima we solve for f'(x)=0 and analyze further.

We have $f(x)= \frac{ x−8}{x+8}=1-\frac{16}{x+8} ​$ ,

differentiating we get

$f'(x)=\frac{16}{(x+8)^2}$

so we can see that $f'(x)$ is a squared quantity which will be positive over all the interval, hence $f(x)$ is an increasing function

and $f'(x)$ has no roots so the maximum and minimum will be positive and negative infinity.

I have tried to show this function by a graph attached below.