Find The Absolute Maximum Value For The Function F(x) = X2 4, On The Interval [3, 0) U (0, 2].

Find The Absolute Maximum Value For The Function F(x) = X2 4, On The Interval [3, 0) U (0, 2]. Determining the absolute maximum of a function over a specified interval is a fundamental concept in calculus, often involving the analysis of critical points and boundary values. In this article, we will explore the process of finding the maximum value of the given function, F(x) = x² + 4, over the combined interval [3, 0) U (0, 2]. We will discuss the method systematically and provide detailed explanations to ensure clarity.

Understanding the Function and the Interval

The Function F(x) = x² + 4

The function in question is a quadratic function:

• Form: F(x) = x² + 4
• Type: Parabola opening upward
• Vertex: At x = 0, since the quadratic term is positive
• Range: For all real x, F(x) ≥ 4

This is a smooth, continuous function with no discontinuities, which simplifies the analysis of its extrema.

Interval Specification and Its Implications

The interval provided is [3, 0) U (0, 2]. Notice:

• It is a union of two parts, excluding the point x=0.
• The first interval is from 3 down to just before 0, i.e., [3, 0).
• The second interval is from just after 0 to 2, i.e., (0, 2].

This structure indicates that the function's maximum might occur at boundary points of these intervals or at critical points within the intervals.

Analyzing the Function on the Specified Intervals

Continuity and Behavior of F(x)

Since F(x) = x² + 4 is continuous everywhere, the maximum value over a closed interval typically occurs at critical points or at the endpoints. However, our interval includes open intervals at 0, so we must consider limits at that point.

Important observations:

• At x=3, F(3) = 3² + 4 = 9 + 4 = 13


• As x approaches 0 from the left, F(x) approaches 4 (since lim x→0− x² + 4 = 4)


• As x approaches 0 from the right, F(x) also approaches 4


• At x=2, F(2) = 4 + 4 = 8

Because the intervals are open at 0, the value of F(x) at x=0 is not included, but the limits as x approaches 0 are relevant for the maximum.

Finding Critical Points

To find potential maximum points within the intervals, we analyze the derivative:

\[ F'(x) = 2x \]

Critical points occur where \( F'(x) = 0 \):

• \( 2x = 0 \Rightarrow x=0 \)

Since x=0 is a critical point, but it's excluded from the interval (open at 0), we must consider the behavior approaching 0 from both sides.

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Determining the Absolute Maximum Value

Step 1: Evaluate Endpoints and Critical Points

• At x=3: F(3)=13
• Approaching x=0 from the left: F(x) approaches 4
• Approaching x=0 from the right: F(x) approaches 4
• At x=2: F(2)=8

Since the point x=0 is not included, the function approaches 4 but never attains it at x=0.

Step 2: Consider the Behavior at the Endpoints of the Intervals

• For [3, 0):
• The maximum value is at x=3, which is 13
• As x approaches 0 from the left, F(x) approaches 4, which is less than 13
• For (0, 2]:
• The maximum value is at x=2, which is 8
• As x approaches 0 from the right, F(x) approaches 4, which is less than 8

Step 3: Summarize the Maximum Values

• On [3, 0): maximum is at x=3, with F(3)=13
• On (0, 2]: maximum is at x=2, with F(2)=8

Since the intervals are disjoint and the function is continuous (except at 0, which is not included), the overall absolute maximum over the combined domain is the highest among these maximums:

\[ \boxed{
\text{Absolute Maximum} = \max\{13, 8\} = 13
} \]

This maximum is attained at x=3.

Conclusion: The Absolute Maximum Value

The absolute maximum value of the function \( F(x) = x^2 + 4 \) on the combined interval \([3, 0) \cup (0, 2]\) is 13, which occurs at x=3. The function approaches 4 near x=0 but does not attain it, and the maximum within the second interval at x=2 is only 8, which is less than 13.

Additional Insights and Tips for Similar Problems

1. Always Check Critical Points and Endpoints

• For continuous functions on closed intervals, potential extrema are at critical points or endpoints.
• For open intervals, consider limits approaching boundary points.

2. Consider the Effect of Open Intervals

• When the interval excludes boundary points, analyze the behavior of the function as it approaches those points.
• Use limits to determine the supremum or infimum.

3. Use Derivatives for Critical Points

• Find where the derivative is zero or undefined to locate potential extrema inside the interval.

4. Compare Values at These Points

• The maximum value is the largest among the critical points and boundary points.

Final Thoughts

Understanding how to analyze functions over complex intervals, especially those involving open and closed parts, is crucial in calculus. By systematically evaluating function values at critical points and boundary limits, you can accurately determine the absolute maximum or minimum values. In this case, recognizing that the maximum occurs at x=3 simplifies the problem, as the function's parabola opens upward and increasing x increases the value of F(x).

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Summary:

• The function \(F(x) = x^2 + 4\) reaches its maximum value of 13 at x=3 within the interval \([3, 0) \cup (0, 2]\).


• The behavior near x=0 approaches 4 but does not include this value, so it cannot be considered the maximum.


• Always analyze critical points and boundary behaviors when dealing with piecewise or union intervals to accurately find extrema.

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