which interval for the graphed function contains the local minimum

Guri Bee

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22-10-2024

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Finding the Local Minimum Interval on a Graph

Understanding how to identify the interval containing a local minimum on a graph is crucial in calculus and other areas of mathematics. It helps us understand the behavior of a function and its critical points. This article will guide you through the process, using examples and explanations based on real-world applications.

What is a Local Minimum?

A local minimum, also known as a relative minimum, is a point on a graph where the function's value is lower than its neighboring points within a certain interval. This means the function "dips" at that point before rising again.

Identifying the Interval Containing the Local Minimum

1. Visual Inspection: Start by visually examining the graph. Look for points where the curve dips and then rises again. This is a good starting point to identify potential local minimums.

2. The First Derivative Test: For a more rigorous approach, use the first derivative test. This involves:

   • Finding the critical points: These are points where the derivative of the function is either zero or undefined.
   • Analyzing the sign of the derivative: To the left of a local minimum, the derivative will be negative (the function is decreasing), and to the right of it, the derivative will be positive (the function is increasing).

Example

Let's consider the function f(x) = x^3 - 3x^2 + 2

1. Visual Inspection:

You can use a graphing calculator or online graphing tool to visualize the function. By observing the graph, you'll notice a dip around x = 1. This suggests a potential local minimum.

2. The First Derivative Test:

• Find the derivative: f'(x) = 3x^2 - 6x
• Set the derivative to zero: 3x^2 - 6x = 0
• Solve for x: x = 0 and x = 2 are the critical points.
• Analyze the sign of the derivative:
  • For x < 0, f'(x) > 0 (increasing)
  • For 0 < x < 2, f'(x) < 0 (decreasing)
  • For x > 2, f'(x) > 0 (increasing)

Therefore, the function has a local minimum at x = 2.

Practical Applications:

• Optimization Problems: In fields like engineering and economics, finding the minimum of a function is crucial for optimizing processes and resources. For example, a company might want to find the minimum cost of producing a certain product.
• Modeling Real-World Phenomena: Many natural phenomena can be modeled using functions. Finding local minima helps us understand the lowest points or optimal states of these phenomena.

Attribution:

• The example function f(x) = x^3 - 3x^2 + 2 is commonly used in calculus examples and is a standard function for demonstrating concepts like local minima.

Note: While the first derivative test is a powerful tool for finding local minima, it's important to consider cases where the second derivative might be helpful for confirming the nature of the critical point.

Conclusion

Identifying the interval containing a local minimum is a fundamental skill in calculus and its applications. By using visual inspection and the first derivative test, you can accurately pinpoint the minimum point of a function and understand its behavior. This knowledge is valuable in various fields, from optimizing processes to modeling real-world phenomena.