Math Problem Statement
Solution
To solve the given problem, we need to analyze the intervals of increase and decrease for the polynomial function .
Step 1: Find the first derivative of
The first derivative, , helps identify critical points and the behavior of :
Step 2: Solve to find critical points
Set : Let , so the equation becomes: Solve using the quadratic formula: Thus, or . This gives:
Step 3: Determine the sign of in each interval
The critical points divide the real number line into intervals: , , , , . Test in each interval:
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Interval : Pick . Substitute into : is increasing.
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Interval : Pick . Substitute into : Calculation shows . is decreasing.
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Interval : Pick . Substitute: is increasing.
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Interval : Pick . Substitute: Calculation shows . is decreasing.
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Interval : Pick . Substitute: is increasing.
Step 4: Write the intervals
- is increasing on .
- is decreasing on .
Would you like a step-by-step walkthrough or further clarification?
Related questions:
- How do we find higher-order derivatives to classify critical points further?
- What role does the second derivative play in determining concavity?
- Can you use a graphing utility to verify these intervals visually?
- How can you confirm the nature of turning points using ?
- What happens to as ?
Tip: Always test critical points carefully to avoid algebraic errors when solving for intervals.
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Math Problem Analysis
Mathematical Concepts
Calculus
Critical Points
Increasing and Decreasing Functions
Formulas
First derivative f'(x)
Quadratic equation formula ax^2 + bx + c = 0
Theorems
Critical point theorem
Sign test for derivative
Suitable Grade Level
Undergraduate - Calculus I
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