Verified Answer :

  • We define a local maximum for a discrete function as a point where the value is greater than or equal to its immediate neighbors.
  • We evaluate the function values \(f(x)\) at each candidate \(x\): \(f(-3) = -2\), \(f(-2) = 0\), and \(f(-1) = 6\).
  • We compare these to their respective neighbors; \(f(-1) = 6\) is greater than its neighbors \(f(-2) = 0\) and \(f(0) = 0\).
  • Therefore, the local maximum occurs at \(x = \boxed{-1}\).

Explanation

  1. Analysis of the Problem

The goal is to identify the x-value where the function \(f(x)\) has a local maximum, given a set of discrete data points in a table. A local maximum for a function defined on a discrete set of points occurs at a value \(x_0\) if the value \(f(x_0)\) is greater than or equal to the function values at the neighboring points in the domain. We examine the points listed in the table: \(x \in \{-4, -3, -2, -1, 0, 1\}\) with corresponding \(f(x) \in \{16, -2, 0, 6, 0, -2\}\). We check the candidate x-values from the options: \(-4, -3, -2, -1\).

  1. Evaluating Candidate Points

A local maximum at \(x_0\) requires \(f(x_0) \geq f(x_0 - 1)\) and \(f(x_0) \geq f(x_0 + 1)\).

  • At \(x = -3\): The neighbors are \(f(-4) = 16\) and \(f(-2) = 0\). Since \(f(-3) = -2\), it is less than both neighbors (specifically \(f(-3) < f(-4)\)), so it is not a local maximum.
  • At \(x = -2\): The neighbors are \(f(-3) = -2\) and \(f(-1) = 6\). Since \(f(-2) = 0\), it is greater than the left neighbor but less than the right neighbor (\(f(-2) < f(-1)\)), so it is not a local maximum.
  • At \(x = -1\): The neighbors are \(f(-2) = 0\) and \(f(0) = 0\). Since \(f(-1) = 6\), it is greater than both its neighbors (\(6 > 0\)), which satisfies the definition of a local maximum for this discrete data set.
  • At \(x = -4\): While the value \(f(-4) = 16\) is high, it is a boundary point. In standard calculus, local maxima are defined for interior points. Even if considered, it lacks a left neighbor to compare against. Given the options, \(-1\) is the clear local maximum.
  1. Final Conclusion

Based on our comparison, the only interior point among the options that satisfies the definition of a local maximum, where the function value is greater than those of its immediate neighbors, is \(x = -1\). Thus, the correct x-value is \(-1\).

Examples

In data science, identifying local maxima is crucial for analyzing trends. For example, if you are tracking the temperature in a room over several hours at discrete time intervals, a local maximum would indicate the peak temperature during a specific window, helping you determine when your heating or cooling system was most active. Similarly, analyzing stock prices at fixed daily intervals can help investors identify "peaks" or local highs in the market's performance, which is exactly how we identified the maximum in this discrete data table.

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