Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.


a. The function f(x) = √x has a local maximum on the interval [0,∞).

Verified step by step guidance

1. To determine if the function f(x) = √x has a local maximum on the interval [0, ∞), we first need to understand what a local maximum is. A function f(x) has a local maximum at a point x = c if there exists an interval (a, b) containing c such that f(c) ≥ f(x) for all x in (a, b).
2. Next, consider the behavior of the function f(x) = √x. As x increases from 0 to ∞, the value of √x also increases. This suggests that the function is monotonically increasing on the interval [0, ∞).
3. To further analyze, we can take the derivative of f(x) to examine its critical points. The derivative f'(x) = (1/2)x^(-1/2) is positive for all x > 0, indicating that the function is increasing on its entire domain.
4. Since the function is increasing and has no points where the derivative is zero or undefined (other than at x = 0, where the derivative is undefined but the function is still increasing), there are no local maxima on the interval [0, ∞).
5. Therefore, the statement that the function f(x) = √x has a local maximum on the interval [0, ∞) is false. The function does not have a local maximum because it continuously increases without reaching a peak.

Key Concepts

Local Maximum

A local maximum of a function occurs at a point where the function's value is greater than the values of the function at nearby points. To determine if a function has a local maximum, one typically examines the first derivative to find critical points and the second derivative to assess concavity. In the context of the given function, understanding local maxima is essential to evaluate its behavior over the specified interval.

Derivative and Critical Points

The derivative of a function provides information about its rate of change and is used to find critical points, where the derivative is zero or undefined. These points are potential candidates for local maxima or minima. For the function f(x) = √x, calculating the derivative helps identify whether there are any points in the interval [0, ∞) where the function reaches a local maximum.

Behavior of the Function on the Interval

Analyzing the behavior of the function on the specified interval is crucial for determining the presence of local maxima. For f(x) = √x, one must consider how the function behaves as x approaches 0 and as x increases towards infinity. Understanding the overall trend of the function helps in concluding whether it can attain a local maximum within the given interval.