- Identify the definition of a local maximum for discrete data: \(f(a-1) < f(a) > f(a+1)\).
- Test each provided point by comparing it to its adjacent values in the table.
- Confirm \((0, 64)\) as the only point satisfying the maximum condition because \(45 < 64 > 45\).
- Final result: \(\boxed{(0, 64)}\).
Explanation
- Analysis of Discrete Data and Local Maximum Criteria
The given function \(f(x)\) is represented by a set of discrete data points: \((-2, 0), (-1, 45), (0, 64), (1, 45), (2, 0), (3, -35), (4, 0), (5, 189), (6, 640)\). A local maximum occurs at a point \(x = a\) if the value of the function at that point is greater than the values of the function at the points immediately preceding and succeeding it in the sequence. That is, we look for \(f(a)\) such that \(f(a-1) < f(a)\) and \(f(a+1) < f(a)\).
- Evaluating Candidate Points
We evaluate the candidate points provided:
- Option \((0, 64)\): The neighbors are \(f(-1) = 45\) and \(f(1) = 45\). Since \(64 > 45\), the function value at \(x=0\) is greater than its immediate neighbors, confirming it is a local maximum.
- Option \((3, -35)\): The neighbors are \(f(2) = 0\) and \(f(4) = 0\). Since \(-35 < 0\), this point is a local minimum, not a maximum.
- Option \((5, 189)\): The neighbors are \(f(4) = 0\) and \(f(6) = 640\). Since \(189 < 640\), the value is increasing at this point, so it is not a local maximum.
- Option \((2, 0)\): The neighbors are \(f(1) = 45\) and \(f(3) = -35\). Since \(0 < 45\), this point is not a local maximum.
- Final Conclusion
Based on the evaluation of the neighbors for each candidate, the only point where the function value is greater than its immediate neighbors is \((0, 64)\). Thus, the local maximum is \(\boxed{(0, 64)}\).
Examples
Understanding how to identify local maxima from discrete data is crucial in fields like data analysis and statistics. For instance, if you are monitoring the daily temperature or the stock market prices, you can use these simple comparisons to identify "peaks" or periods of highest performance. This foundational skill allows you to quickly interpret trends and make data-driven decisions based on observations captured at discrete time intervals.