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Monotonicity Formulas

Monotonicity refers to the behavior of functions in terms of increasing or decreasing intervals based on the first derivative. A function is increasing if its derivative is positive, decreasing if negative, and constant if zero. Critical points are used to determine intervals of monotonicity, which aids in curve sketching and optimization.

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37 views2 pages

Monotonicity Formulas

Monotonicity refers to the behavior of functions in terms of increasing or decreasing intervals based on the first derivative. A function is increasing if its derivative is positive, decreasing if negative, and constant if zero. Critical points are used to determine intervals of monotonicity, which aids in curve sketching and optimization.

Uploaded by

sonuteja13
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© © All Rights Reserved
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Monotonicity - All Important Formulas and Concepts

Monotonicity Basics:
Monotonicity tells us where a function is increasing or decreasing.

1. Definition:
- A function f(x) is:
- Increasing in an interval if f'(x) > 0
- Decreasing in an interval if f'(x) < 0
- Constant if f'(x) = 0

Formulas and Rules:

2. First Derivative Test (Monotonicity):


Used to check where the function is increasing/decreasing.
- If f'(x) > 0 in an interval => f(x) is increasing
- If f'(x) < 0 in an interval => f(x) is decreasing

3. Critical Points:
Points where:
- f'(x) = 0 or
- f'(x) is undefined
Use these to split the domain and test sign of f'(x).

4. Intervals of Monotonicity:
- Find critical points of f'(x)
- Use number line method to test sign of f'(x) in each interval
- Positive => Increasing; Negative => Decreasing

5. Behavior Summary:
- If f'(x) changes from + to - at a point => local maximum
- If f'(x) changes from - to + => local minimum
6. Application:
- Helps in curve sketching, finding turning points, optimization

7. Higher Order Derivatives (Optional):


If f''(x) > 0 => f(x) is concave upward (but not necessarily increasing)
If f''(x) < 0 => f(x) is concave downward

Tip:
Always draw a sign chart of f'(x) to make analysis easier.

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