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).
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
To find the value of A where the function is not defined, we need to identify the value(s) of x that would make the denominator equal to zero. In this case, the denominator is 6x + 3.
Setting the denominator equal to zero, we have:
6x + 3 = 0
Subtracting 3 from both sides, we get:
6x = -3
Dividing both sides by 6, we obtain:
x = -3/6
Simplifying further, we have:
x = -1/2
So, the value of A where the function is not defined is x = -1/2.
Now, let's determine the increasing or decreasing nature of the function within the given intervals: (-∞, A) and (A, ∞).
1. For the interval (-∞, A):
To determine if the function is increasing or decreasing, we need to evaluate the derivative of the function f'(x).
Taking the derivative of f(x) = (2x + 8)/(6x + 3), we get:
f'(x) = (2(6x + 3) - (2x + 8)(6))/(6x + 3)^2
Simplifying the derivative, we have:
f'(x) = (12x + 6 - 12x - 48)/(6x + 3)^2
f'(x) = -42/(6x + 3)^2
Since the derivative f'(x) is negative for all x values in the interval (-∞, A), the function is decreasing in this interval.
2. For the interval (A, ∞):
Again, we evaluate the derivative of f(x) to determine if the function is increasing or decreasing.
Using the same derivative obtained earlier, f'(x) = -42/(6x + 3)^2.
Since the derivative f'(x) is negative for all x values in the interval (A, ∞), the function is also decreasing in this interval.
As mentioned, this function has no inflection points and we can skip the concavity analysis for this particular case.