Determine the intervals on which f(x)=x+(4/x) is increasing and decreasing. Find the coordinates of any max or min points.
To determine the intervals on which the function f(x) = x + (4/x) is increasing or decreasing, we need to analyze the behavior of its derivative, f'(x).
Let's start by finding the derivative:
f'(x) = d/dx (x + (4/x))
= 1 - 4/x^2
Now, to find the intervals on which f(x) is increasing or decreasing, we need to examine the sign of f'(x).
1. Identify the critical points by setting f'(x) = 0.
1 - 4/x^2 = 0
4/x^2 = 1
x^2 = 4
x = Β±2
So, the critical points of f(x) are x = 2 and x = -2.
2. Analyze the sign of f'(x) for different intervals.
For x < -2: Choose a test value, let's say x = -3.
Plug this value into f'(x): 1 - 4/(-3)^2 = 1 - 4/9 = 1 - 4/9 = 5/9 > 0
Since the value of f'(x) is positive, f(x) is increasing for x < -2.
For -2 < x < 2: Choose a test value, let's say x = 0.
Plug this value into f'(x): 1 - 4/0^2 = 1 - undefined = undefined
Since the value of f'(x) is undefined, we cannot determine if f(x) is increasing or decreasing for -2 < x < 2.
For x > 2: Choose a test value, let's say x = 3.
Plug this value into f'(x): 1 - 4/3^2 = 1 - 4/9 = 5/9 > 0
Since the value of f'(x) is positive, f(x) is increasing for x > 2.
3. Now, let's find any possible max or min points. These occur at the critical points where f'(x) changes from positive to negative or vice versa.
At x = 2: Plug x = 2 into f'(x): 1 - 4/2^2 = 1 - 1 = 0
Since the value of f'(x) is 0, this critical point represents a possible maximum point.
The coordinates of this point are (2, f(2)).
At x = -2: Plug x = -2 into f'(x): 1 - 4/(-2)^2 = 1 - 1 = 0
Since the value of f'(x) is 0, this critical point represents a possible minimum point.
The coordinates of this point are (-2, f(-2)).
Therefore, the intervals on which f(x) = x + (4/x) is increasing are: (-β, -2) and (2, +β).
The intervals where f(x) might be decreasing or have local minima are: (-2, 2).
The maximum point is located at (2, f(2)), and the minimum point is located at (-2, f(-2)).