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 and decreasing, we need to find the derivative of the function first. The derivative will give us information about the slope of the function at each point.
1. Find the derivative of f(x):
Using the power rule and quotient rule, the derivative of f(x) is:
f'(x) = 1 - 4/x^2
2. Set f'(x) equal to zero to find critical points:
1 - 4/x^2 = 0
Solve for x:
4/x^2 = 1
4 = x^2
x = Β±2
Therefore, we have two critical points at x = -2 and x = 2.
3. Determine the intervals:
To determine the intervals of increasing and decreasing, we can use the test points within each interval and evaluate the derivative.
a) For x < -2:
Choose x = -3 as the test point.
Substitute x = -3 into f'(x) to find the sign: f'(-3) = 1 - 4/(-3)^2 = 1 - 4/9 = 5/9
Since f'(-3) > 0, the function is increasing for x < -2.
b) For -2 < x < 2:
Choose x = 0 as the test point.
Substitute x = 0 into f'(x) to find the sign: f'(0) = 1 - 4/0^2 = undefined
Since f'(0) is undefined, we need to check the behavior around x = -2 and x = 2.
Choose x = -1 as a test point before x = -2:
Substitute x = -1 into f'(x) to find the sign: f'(-1) = 1 - 4/(-1)^2 = -3
Since f'(-1) < 0, the function is decreasing for -2 < x < 2 (before passing x = -2).
Choose x = 1 as a test point after x = -2:
Substitute x = 1 into f'(x) to find the sign: f'(1) = 1 - 4/1^2 = -3
Since f'(1) < 0, the function is still decreasing for -2 < x < 2 (before reaching x = 2).
c) For x > 2:
Choose x = 3 as the test point.
Substitute x = 3 into f'(x) to find the sign: f'(3) = 1 - 4/3^2 = 1 - 4/9 = 5/9
Since f'(3) > 0, the function is increasing for x > 2.
4. Determine the max and min points:
The max and min points occur at the critical points where the derivative changes sign. So, we need to further investigate the points x = -2 and x = 2.
- When x = -2:
Substitute x = -2 into the original function f(x): f(-2) = -2 + (4/-2) = -2 - 2 = -4
The coordinates of the minimum point are (-2, -4).
- When x = 2:
Substitute x = 2 into the original function f(x): f(2) = 2 + (4/2) = 2 + 2 = 4
The coordinates of the maximum point are (2, 4).
Therefore, the function f(x) = x + (4/x) is increasing for x < -2 and x > 2, and it is decreasing for -2 < x < 2. The coordinates of the minimum point are (-2, -4), and the coordinates of the maximum point are (2, 4).