Math Problem Statement
f(x)= x^2-6x+8; a=1, b=3
Solution
Given the function , we are interested in analyzing the behavior of the function over the interval from to .
Let's go step by step:
- Find the value of at the endpoints:
So, and .
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Find the derivative of to check for critical points: The derivative of is: Setting the derivative equal to zero to find critical points:
The critical point occurs at , but since this is an endpoint of the interval, we don't have any additional critical points inside the interval .
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Analyze the behavior of over the interval:
- At , the function value is 3.
- At , the function value is -1.
The function decreases from to .
Would you like to explore more details on this or perform further calculations?
Here are some follow-up questions:
- What is the vertex of the parabola represented by ?
- How does the function behave beyond the interval ?
- What are the roots of the function ?
- Can you rewrite in vertex form?
- How can we determine if the function has a maximum or minimum within the interval?
Tip: Finding the vertex of a quadratic function helps you understand where the function reaches its maximum or minimum value!
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Math Problem Analysis
Mathematical Concepts
Quadratic Functions
Derivatives
Critical Points
Interval Analysis
Formulas
Quadratic function: f(x) = ax^2 + bx + c
Derivative of a quadratic function: f'(x) = 2ax + b
Theorems
Critical point occurs where the derivative is zero
Quadratic functions have a vertex which represents the maximum or minimum value
Suitable Grade Level
Grades 9-11
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