Problem 64 - Calculus of a Single Variable

Question

True or False? In Exercises \(63-66\) , determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If a function is continuous on a closed interval, then it must have a minimum on the interval.

Step-by-Step Solution

Answer

True. If a function is continuous on a closed interval, then it must have a minimum on the interval.

Step 1: Understanding the statement and terms

A function being continuous on a closed interval means that for any value within that interval, there exists a point on the function graph for that value. The statement further suggests that such a function always have a minimum or the lowest point on the interval.

Step 2: Evaluate the statement

The statement is indeed true. This is a conclusion of the Extreme Value Theorem, which states that if a function is continuous on a closed interval, then the function reaches both a minimum and a maximum value on that interval.

Step 3: Provide an Example

Consider the function \(f(x) = x^2\) on the interval \([-1,1]\). The function is continuous on this interval. The minimum value of the function is 0, which is reached at x=0. Thus, this example shows that if a function is continuously defined on a closed interval, it must attain its minimum in that interval.

Key Concepts

Continuous Function

Understanding what a continuous function is will help us unravel the logic behind the Extreme Value Theorem. In the realm of mathematics, a continuous function is one without breaks, jumps, or holes in its domain. For students, you can think of it like a smooth drawing that you can create without lifting your pencil off the paper.

More formally, a function is considered continuous at a point if the following three conditions are met: 1) the function is defined at that point, 2) the limit of the function as it approaches that point exists, and 3) the limit equals the function's value at that point. When these conditions are satisfied across all points within a certain interval, we describe the function as being continuous on that interval.

For example, the simple function of a line rising steadily, given by the equation \( f(x) = x \), is continuous because at every point along the line, the function smoothly progresses without any interruption.

Closed Interval

A closed interval in mathematics is a range of values that includes its end points. It is typically denoted with square brackets, such as \( [a, b] \), where 'a' and 'b' are the endpoints of the interval. This means that both 'a' and 'b' are part of the interval. Closed intervals are very important in calculus because they allow the application of certain theorems that do not work on open intervals, where the endpoints are not included.

In the context of continuous functions, closed intervals play a critical role. They are like a contained playground, where the behavior of the continuous function is analyzed from one boundary to the other without leaving anything out. The certainty of having both endpoints in the interval ensures that we can always find a maximum and a minimum value, as the play area is well-defined and fenced in, so to speak.

Minimum Value

When students do calculations or graph functions, finding the minimum value is a common task. It is the lowest point on the graph of a function within a certain interval. This 'lowest point' refers to the y-value, not the x-value. In a real-world example, think of the minimum value as the base level of water in a bathtub—the lowest depth at which water sits.

According to the Extreme Value Theorem, a continuous function on a closed interval will always have a minimum value. This makes sense intuitively if you consider that the continuous curve must enter and exit the closed interval it occupies; somewhere in between, it must reach its lowest height. For students visualizing this, imagine walking along a path that smoothly rises and falls; there has to be a lowest point on your walk where the ground is closest to sea level.