Math Problem Statement
Solution
The question asks whether the Intermediate Value Theorem (IVT) applies to the given function on the interval , and if so, to find such that . The possible answers involve whether the theorem applies and the value of .
To apply the IVT:
- Conditions for IVT:
- The function must be continuous on the closed interval .
- If the function is continuous, the IVT states that for any value between and , there exists some such that equals that value.
Step-by-step verification:
-
Check continuity: The function is a quadratic function, which is continuous everywhere, so it is continuous on .
-
Evaluate and :
-
Check if 2 is between and :
- and . Since lies between and , the IVT guarantees that there is some such that .
Find such that :
Solve : Multiply the equation by to simplify: Now solve the quadratic equation using the quadratic formula: Thus, or .
Since must be in the interval , the solution is .
Conclusion:
The Intermediate Value Theorem applies, and the value of is , matching the third option.
Do you have any questions or would you like more details?
Follow-up Questions:
- How does the Intermediate Value Theorem work for more complex functions?
- Why is continuity a critical requirement for the IVT?
- Can the IVT be applied if the function is discontinuous?
- How would you solve this problem graphically?
- What other mathematical theorems are closely related to the IVT?
Tip: When solving quadratic equations, always check the discriminant to see if real solutions exist before applying the quadratic formula!
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Math Problem Analysis
Mathematical Concepts
Calculus
Intermediate Value Theorem
Continuity
Formulas
Quadratic formula: x = (-b ± sqrt(b^2 - 4ac)) / 2a
Theorems
Intermediate Value Theorem
Suitable Grade Level
College Level (Calculus I)
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