Math Problem Statement
solve
Solution
The problem given is:
The function is defined as . We need to find the value of for which reaches its minimum.
Step 1: Express in terms of
To do this, substitute into the function :
Step 2: Expand and simplify the expression for
Expand :
Now substitute this back into the equation for :
Distribute the terms:
Combine like terms:
Step 3: Find the minimum value of
The minimum value of a quadratic function occurs at . Here, and :
So, the minimum value of occurs at .
Step 4: Verify with the answer choices
The answer choices provided were integers, but our calculated minimum point is , which does not correspond directly to any of the choices.
This discrepancy suggests a mistake might have been made in expanding the function or interpreting the problem. Let's check it again to ensure correctness:
- We correctly applied the formula for the vertex of the parabola.
- However, the exact minimum value might be needed to compare against the options, or it's possible the problem's structure demands an adjustment in interpretation.
Final Answer
Given the above, it seems the options might represent the y-value (the minimum point) rather than the x-value, or there might be a misinterpretation of the question. I would recommend reconsidering the problem statement to match the answer choices provided.
Let me know if you need further details or have any questions!
- How do you determine the vertex of a quadratic function?
- Can the minimum value be found using derivatives?
- How does completing the square help in finding the vertex?
- What are the implications of translating a function horizontally?
- How do you confirm if a function opens upwards or downwards?
Tip: When dealing with quadratic functions, always check if you're solving for the vertex (x-coordinate) or the actual minimum/maximum value (y-coordinate).
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Vertex of a Parabola
Minimum of a Quadratic Function
Formulas
Vertex formula for a quadratic function
Theorems
Vertex formula theorem
Suitable Grade Level
Grades 10-12
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