Math Problem Statement
Solution
Let's work through the problem step by step for the function .
1. Expanding the Function
First, expand the function :
Notice that is a difference of squares, so:
Or if expanded completely:
Expanding this:
2. First Derivative
Differentiate with respect to :
The derivatives of each term are:
3. Second Derivative
Differentiate to find the second derivative:
4. Stationary Points
Stationary points occur where . Set :
Divide the entire equation by 3:
Factorize the quadratic equation:
Thus, and are the stationary points.
5. Nature of Stationary Points
Evaluate the second derivative at the stationary points to determine their nature:
- For :
Since , is a point of minimum.
- For :
Since , is a point of maximum.
6. Sketching the Graph
The graph will have:
- A minimum point at
- A maximum point at
To find the corresponding -values, substitute these -values back into the original function .
So:
- Minimum at
- Maximum at
Would you like any further details or a deeper explanation on any part of this solution?
Here are 5 questions to further explore:
- How does factorizing the expression help in determining the roots and stationary points?
- Why is the second derivative useful in determining the nature of stationary points?
- What would be the effect on the graph if the constant term in the expanded form was different?
- How can we use the first derivative test instead of the second derivative test to determine the nature of stationary points?
- How would the graph behave at very large positive and negative values of ?
Tip:
When expanding or factoring polynomials, always recheck each step to avoid small errors that can lead to incorrect stationary points or graph shapes.
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Math Problem Analysis
Mathematical Concepts
Polynomials
Quadratic Equations
Derivatives
Stationary Points
Formulas
Difference of squares
Quadratic formula
Theorems
Second derivative test
Suitable Grade Level
Advanced High School
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