A cost function is given by: A'(x)= 2x^2/x^2-211/x^2.
For what values is A(x) increasing and decreasing?
Any help would be appreciated
To determine where the function A(x) is increasing or decreasing, we need to find the critical points. These are the values of x where the derivative of A(x) equals zero or is undefined. In other words, we want to find x such that A'(x) = 0 or A'(x) is undefined.
First, let's find A'(x):
A'(x) = 2x^2/(x^2 - 211/x^2)
To find the critical points, we need to set A'(x) = 0 and solve for x:
2x^2/(x^2 - 211/x^2) = 0
To solve this equation, recall that a fraction is only equal to zero if the numerator is zero. So, we set the numerator equal to zero:
2x^2 = 0
Solving for x, we find that x = 0.
Now, let's check if A'(x) is undefined for any values of x. In our case, A'(x) is undefined when the denominator is zero:
x^2 - 211/x^2 = 0
To solve this equation, multiply both sides by x^2 to eliminate the denominator:
x^4 - 211 = 0
Now, we have a fourth-degree polynomial equation. Unfortunately, it does not factor nicely. We can use numerical methods or a calculator to find the solutions:
x = ±√(211)
So, the critical points are x = -√211, 0, and √211.
To determine where A(x) is increasing or decreasing, we'll use the first derivative test. The first derivative test states that if the derivative of a function changes sign at a critical point, then the function has a local extremum (maximum or minimum) at that point. If the derivative does not change sign, then the function is either always increasing or always decreasing on that interval.
Let's examine the intervals created by the critical points:
1. Interval (-∞, -√211):
We need to choose a test point within this interval. Let's choose x = -1. Plugging this into A'(x), we get:
A'(-1) = 2(-1)^2/((-1)^2 - 211/(-1)^2) = 2/212
Since A'(-1) is positive, A(x) is increasing on this interval.
2. Interval (-√211, 0):
Again, we need a test point within this interval. Let's choose x = -0.5. Plugging this into A'(x), we get:
A'(-0.5) = 2(-0.5)^2/((-0.5)^2 - 211/(-0.5)^2) = 0
Since A'(-0.5) equals zero, we cannot determine the behavior of A(x) on this interval using the first derivative test.
3. Interval (0, √211):
For this interval, let's choose x = 0.5 as our test point:
A'(0.5) = 2(0.5)^2/((0.5)^2 - 211/(0.5)^2) = 0
Again, A'(0.5) equals zero, so we cannot determine the behavior of A(x) on this interval using the first derivative test.
4. Interval (√211, ∞):
Let's choose x = √211 + 1 as our test point within this interval:
A'(√211 + 1) = 2(√211 + 1)^2/((√211 + 1)^2 - 211/((√211 + 1))^2) = -2/212
Since A'(√211 + 1) is negative, A(x) is decreasing on this interval.
Therefore, A(x) is increasing on (-∞, -√211) and decreasing on (√211, ∞). On the interval (-√211, 0) and (0, √211), the behavior of A(x) cannot be determined using the first derivative test.