Math Problem Statement
Solution
The function given is , and the goal is to find the range of values of for which given that .
Step 1: Factorize the quadratic expression
First, we factor the quadratic expression .
We can factor it as:
So, the function becomes:
Step 2: Find the roots of the function
The function will be zero when any of the factors are zero. The roots are:
These roots divide the real line into different intervals. We are interested in the behavior of the function within the interval .
Step 3: Analyze the sign of the function in
We now check the sign of within the given interval:
- At , (positive).
- At , .
- At , .
Let's calculate :
which is negative.
Step 4: Conclusion
The function between and . For values of in , the function becomes negative.
Thus, the range of values of for which in the interval is:
Would you like further details on any step? Here are five related questions to expand on this:
- How can you verify the behavior of for other intervals outside ?
- What role do critical points play in finding the sign of a function?
- How do you determine the concavity of a function like ?
- Can you extend this problem to find where ?
- How would the analysis change if the interval for were different?
Tip: When solving inequalities involving factored polynomials, it is useful to evaluate the sign of each factor across the intervals formed by the roots.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Inequalities
Formulas
Quadratic Factorization
Polynomial Sign Analysis
Theorems
Zero Product Property
Sign Analysis for Polynomials
Suitable Grade Level
Grades 9-11