determine the domain on which the following function is decreasing

Guri Bee

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20-10-2024

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Determining Where a Function Decreases: A Step-by-Step Guide

Understanding where a function decreases is crucial in calculus and various applications. This article will guide you through the process of determining the decreasing intervals of a function, using practical examples and insights from insightful discussions on GitHub.

The Basics: What Does it Mean for a Function to Decrease?

A function is considered decreasing on an interval if its output values (y-values) become smaller as the input values (x-values) increase. Visually, the graph of the function will slope downward as you move from left to right.

Key Tools: Derivatives to the Rescue

The core tool for determining where a function decreases is the derivative. The derivative of a function tells us the instantaneous rate of change at any point.

• Positive derivative: The function is increasing.
• Negative derivative: The function is decreasing.
• Zero derivative: The function might have a local maximum or minimum.

Step-by-Step Process

1. Find the derivative: Calculate the derivative of the function using the appropriate rules of differentiation.
2. Find critical points: Set the derivative equal to zero and solve for x. These points are called critical points.
3. Test intervals: Divide the x-axis into intervals using the critical points. Choose a test point within each interval and evaluate the derivative at that point. If the derivative is negative, the function is decreasing in that interval.
4. Write the decreasing interval: Write down the intervals where the derivative is negative.

Example: Analyzing the Function f(x) = x^3 - 3x^2

Let's illustrate the process using the function f(x) = x^3 - 3x^2.

1. Derivative: f'(x) = 3x^2 - 6x
2. Critical points: 3x^2 - 6x = 0 => 3x(x - 2) = 0 => x = 0, x = 2
3. Test intervals:
   • x < 0: Choose x = -1. f'(-1) = 9 > 0 (Increasing)
   • 0 < x < 2: Choose x = 1. f'(1) = -3 < 0 (Decreasing)
   • x > 2: Choose x = 3. f'(3) = 9 > 0 (Increasing)
4. Decreasing interval: The function is decreasing on the interval (0, 2).

Additional Considerations

• Endpoint behavior: Pay attention to the behavior of the function at the endpoints of the interval. If the function approaches infinity or negative infinity, it might be decreasing without a specific interval.
• Discontinuities: If the function has any discontinuities, the intervals of decrease might be affected.
• Higher-order derivatives: For more complex functions, you might need to analyze higher-order derivatives to determine the intervals of decrease.

GitHub Insights

On GitHub, discussions regarding determining decreasing intervals often revolve around specific functions, challenging scenarios, and alternative approaches. For instance, some users might focus on using the first derivative test, while others prefer the second derivative test for more accurate analysis. These discussions provide valuable perspectives and alternative strategies for tackling different types of functions.

Conclusion

By understanding the relationship between the derivative and the decreasing behavior of a function, you can effectively determine the intervals where a function decreases. This process is essential for optimization problems, curve sketching, and understanding the behavior of functions in various fields. Remember to always analyze the endpoints, discontinuities, and use higher-order derivatives if necessary for a comprehensive analysis.