Question

Multiple Choice A function that is continuous on the interval ________________ is guaranteed to have both an absolute maximum and an absolute minimum. (a) \((a, b)\) (b) \((a, b]\) (c) \([a, b)\) (d) \([a, b]\)

Short answer

(d) ([a, b])

Step-by-step solution

- Understand the Problem

Identify what the question is asking. The problem is asking for the type of interval on which a continuous function is guaranteed to have both an absolute maximum and an absolute minimum.

- Remember the Extreme Value Theorem

Recall the Extreme Value Theorem from calculus. The theorem states that if a function is continuous on a closed interval ([a, b]), then the function must attain an absolute maximum and an absolute minimum on that interval.

- Analyze the Interval Options

Examine each of the interval options provided:(a) ((a, b)) - Open interval, not closed(b) ((a, b]) - Half-open interval(c) ([a, b)) - Half-open interval(d) ([a, b]) - Closed interval

- Choose the Correct Interval

Based on the Extreme Value Theorem, the correct interval is the one that is closed on both ends: ([a, b]).

Key concepts

continuous function

Imagine a graph that you can draw without lifting your pencil off the paper. This is what a continuous function looks like. All points on the graph are connected smoothly, with no breaks or jumps. The idea is that for every point in the interval, the function has a value.

A continuous function is important because it ensures that there are no gaps in the graph. If the function was not continuous, it would be difficult to determine if an absolute maximum or minimum exists, since there could be sudden jumps or undefined points.

closed interval

A closed interval includes all its endpoints and is written as \([a, b]\). It means that the values at 'a' and 'b' are included in the set. This is unlike an open interval which does not include its endpoints, written as (a, b). For instance, in a closed interval \([2, 5]\), the values 2 and 5 are part of the interval. However, in an open interval (2, 5), 2 and 5 are not included, only values between them.

Understanding the difference between closed and open intervals is crucial for applying the Extreme Value Theorem properly. The theorem specifically requires a closed interval \([a, b]\) to guarantee the existence of absolute maximum and minimum values.

absolute maximum

The absolute maximum of a function on a closed interval \([a, b]\) is the highest value the function reaches on that interval. In other words, it's the peak or the highest point on the graph between 'a' and 'b'.

You can think of it as the tallest hill in a landscape. For example, if you have a function f(x) that is continuous on the interval \([1, 4]\) and the highest point it reaches is 10, then 10 is the absolute maximum on \([1, 4]\). The Extreme Value Theorem guarantees that if the function is continuous on the closed interval, it will have such a maximum value.

absolute minimum

The absolute minimum of a function on a closed interval \([a, b]\) is the lowest value the function reaches on that interval. It's like the deepest valley or the lowest point on the graph between 'a' and 'b'.

For instance, if you have a function f(x) that is continuous on the interval \([1, 4]\) and the lowest point it reaches is -3, then -3 is the absolute minimum on \([1, 4]\). According to the Extreme Value Theorem, if the function is continuous on the closed interval, it will have such a minimum value.

Ensuring that both the absolute maximum and absolute minimum exist within the given interval helps us understand the range of the function fully.