f(x)=2x+8x^-1.For this function there are four important intervals: (-infinity,A],[A,B),(B,C), and [C,infinity) where A,B and C are either critical numbers or points at which f(x) is undefined.Find A,B and C.
f'(x) = 2 - 8/x^2
setting this equal to zero yields x = ±2
where y is then ±8
I will describe the function:
Vertical asymptote at x = 0
Minimum point at (2,8)
Maximum point at (-2,-8)
Oblique asymptote y = 2x
To find the critical numbers or points where the function is undefined, we first need to determine where the function is defined and continuous.
Looking at the given function, f(x) = 2x + 8x^(-1), we can see that it consists of two terms: 2x and 8x^(-1), where x^(-1) represents the reciprocal of x.
Now, in order to find the intervals where the function is defined, we need to consider two cases:
1. Case 1: The function is defined for all real numbers except when x = 0 because dividing by zero is undefined.
2. Case 2: The function is defined for all real numbers except when x = 0, as the reciprocal of x is undefined when x = 0.
Now, let's find the critical numbers and points where the function is undefined in each of the given intervals:
1. In the interval (-∞, A]:
Since there are no restrictions or undefined points between -∞ and A, there are no critical numbers or points where the function is undefined in this interval.
2. In the interval [A, B):
In this interval, the critical point will be x = 0 because the function is undefined at x = 0.
3. In the interval (B, C):
Similarly to the previous interval, the critical point will be x = 0.
4. In the interval [C, ∞):
Once again, the critical point is x = 0.
To summarize:
- A critical point or point where the function is undefined is x = 0.
- The intervals where these points occur are [A, B), (B, C), and [C, ∞).
It's important to note that the values of A, B, and C were not provided in the question, so we cannot determine their specific values without additional information.