Math Problem Statement
What ordered pair is closest to a local minimum of the function, f(x)?
Solution
The question is asking for the ordered pair closest to a local minimum of the function , given a table of values.
Table:
x & f(x) \\ \hline -2 & -8 \\ -1 & -3 \\ 0 & -2 \\ 1 & 4 \\ 2 & 1 \\ 3 & 3 \\ \end{array}$$ ### What is a local minimum? A **local minimum** occurs when the value of the function is lower than the values immediately surrounding it. Specifically, if at some $$x = a$$, the function satisfies $$f(a) < f(a-1)$$ and $$f(a) < f(a+1)$$, then $$f(a)$$ is a local minimum. ### Analysis: - At $$x = -2$$, $$f(x) = -8$$. The next point is $$f(-1) = -3$$, which is greater than $$-8$$. This suggests a minimum at $$x = -2$$, but it’s not local since there's no left neighbor to compare. - At $$x = -1$$, $$f(x) = -3$$, and $$f(-2) = -8$$ and $$f(0) = -2$$, so it is not a minimum. - At $$x = 0$$, $$f(x) = -2$$, and $$f(-1) = -3$$, $$f(1) = 4$$, so $$f(0) = -2$$ is larger than the point before it, not a minimum. - At $$x = 1$$, $$f(x) = 4$$, and $$f(0) = -2$$, $$f(2) = 1$$, so it's not a minimum. - At $$x = 2$$, $$f(x) = 1$$, and $$f(1) = 4$$, $$f(3) = 3$$, which is less than both surrounding points. This suggests a local minimum at $$x = 2$$. ### Conclusion: The ordered pair closest to a local minimum is $$(2, 1)$$. Would you like further details on the concept of local minima or any other related topics? Here are 5 related questions that might interest you: 1. What is the difference between a local minimum and a global minimum? 2. How do you determine local minima from a function's derivative? 3. Can a function have more than one local minimum? 4. What is the significance of second derivatives in identifying local minima? 5. How do local minima relate to optimization problems? **Tip:** When analyzing data points for local minima or maxima, always check the values immediately surrounding a point to see if they increase or decrease.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Local minimum
Function analysis
Data interpretation
Formulas
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Theorems
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Suitable Grade Level
Grades 9-11