Question

True or False A function \(f\) has a local minimum at \(c\) if there is an open interval \(I\) containing \(c\) so that \(f(c) \leq f(x)\) for all \(x\) in this open interval.

Short answer

True

Step-by-step solution

- Understand the Definition of Local Minimum

A function has a local minimum at a point if, within some open interval around this point, the function's value at the point is less than or equal to the function's values at all other points in this interval.

- Analyze the Given Statement

The statement says a function f has a local minimum at c if there is an open interval I containing c so that f(c) ≤ f(x) for all x in this open interval.

- Match the Definition to the Statement

Compare this statement to the definition of a local minimum. They both state that within an open interval containing the point c, the value of the function at c is less than or equal to the values of the function at all other points in this interval.

- Conclusion

Since the provided statement matches the definition of a local minimum exactly, the statement is true.

Key concepts

Function Analysis

Function analysis is a crucial part of understanding mathematical concepts. It involves breaking down a function to study its behavior, such as its increasing or decreasing intervals, maxima, and minima.
In this context, we focus on the local minimum, which refers to the point in a function where the value is smallest compared to its neighboring points. Analyzing functions entails:

• Identifying critical points
• Checking the behavior of the function around these points
• Using derivatives to determine where these critical points occur

This kind of analysis helps you understand the different characteristics of functions and solve related problems.

Open Interval

An open interval is a set of numbers between two endpoints, where the endpoints are not included.
In mathematical notation, an open interval from a to b is written as \(a, b\), meaning that if you have an open interval \(I\) containing a point \(c\), then \forall x : c - \epsilon < x < c + \epsilon, \forall \epsilon \in \mathbb{R}^{+}\.
This concept is crucial for discussing local minima because it defines the range of values around a point where the function's behavior is analyzed. If \(c\) is in the open interval and \(f(c) \leq f(x)\) for all \(x\) in this interval, then \(c\) is indeed a local minimum.

Inequality in Functions

Inequality in functions is used to compare values of a function at different points. For a local minimum, the inequality \(f(c) \leq f(x)\) must hold within an open interval.
This means that at the local minimum point \(c\), the function’s value does not exceed the values at nearby points. Here are the key takeaways:

• Understanding \(f(c) \leq f(x)\): If this holds in the interval, then \(c\) is a local minimum
• Applying derivatives: First and second derivatives help confirm these inequalities by indicating the slope and concavity

Grasping this inequality helps in precisely determining local minima and making sound mathematical arguments.

Mathematical Definitions

Mathematical definitions provide the foundation for understanding and solving problems. Let’s break down the definition of a local minimum.
According to the definition, a function \(f\) has a local minimum at \(c\) if there exists an open interval \((I)\) containing \(c\) such that \(f(c) \leq f(x)\) for all \(x\) in the interval. This implies:

• \