Final answer:
The function f(x) = x² increases on the interval (0, ∞) and decreases on the interval (-∞, 0). This is determined by calculating the derivative, f'(x) = 2x, and assessing where the derivative is positive or negative.
Step-by-step explanation:
The intervals of increase and decrease of the function f(x) = x² can be determined by looking at the behavior of the graph of the function. To find these intervals, we differentiate the function to get f'(x) = 2x, which tells us the slope of the tangent line at any point. The function is increasing when the slope is positive (f'(x) > 0) and decreasing when the slope is negative (f'(x) < 0).
From the derivative, we can see that:
- For x > 0, f'(x) = 2x is positive, so the function increases.
- For x < 0, f'(x) = 2x is negative, so the function decreases.
- At x = 0, the function does not increase or decrease, as this is where the minimum point of the quadratic function occurs.
Therefore, the intervals of increase and decrease for f(x) = x² are:
- Increase: (0, ∞)
- Decrease: (-∞, 0)
By comparing the options, we can see that option (b) Increase: (0, ∞), Decrease: (-∞, 0) correctly describes the behavior of f(x) = x².