Question

Consider the function f(x)= (2x+8)/(6x+3). For this function there are two important intervals: (-inf, A)
(A, inf) and where the function is not defined at A .

Find A

For each of the following intervals, tell whether is increasing (type in INC) or decreasing (type in DEC).

Note that this function has no inflection points, but we can still consider its concavity. For each of the following intervals, tell whether is concave up (type in CU) or concave down (type in CD).

Answer 1

f(x) = inf when 6x+3 = 0 or when x=-1/2

That is one of the asymptotes which separate the graph.

as for increasing and decreasing,
well, from -inf going to -1/2
as x increases y decreases

and

from -1/2 to inf
as x increases y decreases also.

If you draw it it has the shape of y = (1/x).

Now for concavity we need to get f''(0) and see if it's +ve or -ve.
It turns out to be +56/3 which means that that is a local max which means that it is decreasing either side - concave down. (-1/2 to inf). By contrast the other half of the graph must then be concave up (-inf to -1/2).

I hope that helps