Let f(x) = 3+4x-x^{3}. Find the open intervals on which f is increasing (decreasing). Then determine the x-coordinates of all relative maxima (minima).
1. f is increasing on the intervals
2. f is decreasing on the intervals
To determine the open intervals on which the function f(x) = 3+4x-x^{3} is increasing or decreasing, we need to find the first derivative of f(x) and analyze its sign.
Step 1: Find the first derivative of f(x)
The derivative of f(x) is found by taking the derivative term by term:
f'(x) = d/dx (3+4x-x^{3})
= 0 + 4 - 3x^{2}
= 4 - 3x^{2}
Step 2: Solve for the critical points
To find the critical points, set the derivative equal to zero and solve for x:
4 - 3x^{2} = 0
3x^{2} = 4
x^{2} = 4/3
x = ±√(4/3)
x = ±2/√3
Step 3: Analyze the sign of the derivative
To determine the intervals where f(x) is increasing or decreasing, we need to analyze the sign of the derivative in each interval.
Interval 1: (-∞, -2/√3)
Choose a value within the interval, let's say x = -1. Substitute this value into the derivative:
f'(-1) = 4 - 3(-1)^{2}
= 4 - 3(1)
= 1
Since the derivative is positive in this interval, f(x) is increasing.
Interval 2: (-2/√3, 2/√3)
Choose a value within the interval, let's say x = 0. Substitute this value into the derivative:
f'(0) = 4 - 3(0)^{2}
= 4 - 0
= 4
Since the derivative is positive in this interval, f(x) is increasing.
Interval 3: (2/√3, ∞)
Choose a value within the interval, let's say x = 1. Substitute this value into the derivative:
f'(1) = 4 - 3(1)^{2}
= 4 - 3(1)
= 1
Since the derivative is positive in this interval, f(x) is increasing.
Therefore, f(x) is increasing on the intervals (-∞, -2/√3), (-2/√3, 2/√3), and (2/√3, ∞).
To find the intervals where f(x) is decreasing, we look at the intervals between the intervals where f(x) is increasing:
Interval 1: (-∞, -2/√3)
The previous interval stopped at -2/√3, so it is decreasing before that point.
Interval 2: (-2/√3, 2/√3)
The next interval starts at 2/√3, so it is decreasing after that point.
Interval 3: (2/√3, ∞)
This is the last interval, so it is decreasing after this point.
Therefore, f(x) is decreasing on the intervals (-∞, -2/√3), (-2/√3, 2/√3), and (2/√3, ∞).
To determine the x-coordinates of all relative maxima and minima, we need to find the second derivative of f(x) and analyze its sign.
Step 1: Find the second derivative of f(x)
The second derivative of f(x) is found by taking the derivative of the first derivative:
f''(x) = d/dx (4 - 3x^{2})
= 0 - 6x
= -6x
Step 2: Set the second derivative equal to zero and solve for x
-6x = 0
x = 0
Step 3: Analyze the sign of the second derivative
To determine if x = 0 is a relative maximum or minimum, we need to analyze the sign of the second derivative.
Choose a value to the left of x = 0, let's say x = -1:
f''(-1) = -6(-1)
= 6
Since the second derivative is positive, x = 0 is a relative minimum.
Choose a value to the right of x = 0, let's say x = 1:
f''(1) = -6(1)
= -6
Since the second derivative is negative, x = 0 is a relative maximum.
Therefore, the x-coordinate of the relative maximum is x = 0, and the x-coordinate of the relative minimum is also x = 0.