True or False: Continuity and Global Minimum

When discussing the implications of continuity on a closed interval, it's essential to understand the key concepts involved. This examination focuses on the statement:

"If a function is continuous on [a, b] then a global minimum is found in c where ( a \leq c \leq b )."

Understanding Continuity

A function ( f(x) ) is said to be continuous on an interval [a, b] if for every ( x ) in the interval, the following three conditions hold:

1. ( f(x) ) is defined at every point in the interval.
2. The limit of ( f(x) ) as ( x ) approaches any point ( c ) within the interval exists.
3. The value of the function at that point equals the limit, ( f(c) = \lim_{x \to c} f(x) ).

Continuity is critical because it implies that there are no jumps, breaks, or asymptotic behaviors in the interval, allowing for a more predictable behavior of the function.

Extreme Value Theorem

The guaranteed existence of global minima and maxima for continuous functions on closed intervals is a crucial result in calculus known as the Extreme Value Theorem. This theorem states:

• If a function ( f ) is continuous on a closed interval [a, b], then ( f ) attains a maximum value and a minimum value at least once in that interval.

Implications of the Theorem

From the Extreme Value Theorem, we can conclude the following:

• Existence of a Global Minimum: There exists at least one point ( c ) in the interval [a, b] such that ( f(c) \leq f(x) ) for all ( x ) in [a, b]. Therefore, this ( c ) is the point where the function achieves its global minimum.
• Location of c: The point ( c ) is guaranteed to be within the interval, so ( a \leq c \leq b ).

Conclusion

Given these established principles, we can affirm the truth of the original statement:

True. If a function is continuous on the interval [a, b], then it is guaranteed to achieve a global minimum somewhere within that interval. The existence of the global minimum is not just a theoretical concept; it is backed by rigorous mathematical proof through the framework of the Extreme Value Theorem.

Summary

To summarize, the assertion that a continuous function on a closed interval must have a global minimum within that interval is a fundamental principle in calculus. It underscores the importance of continuity and the behavior of functions across defined ranges. Students and professionals utilizing these concepts should rely on the Extreme Value Theorem as a foundational theorem in their studies and applications, ensuring that they apply these principles correctly in a wide variety of mathematical contexts.