Problem 58 - Essential Calculus

Question

True or False? In Exercises 55-60, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If \(f\) is increasing on \([a, b],\) then the minimum value of \(f(x)\) on \([a, b]\) is \(f(a)\)

Step-by-Step Solution

Answer

The statement is True, assuming the function is defined and continuous on the interval [a, b].

Step 1: Understand the statement

The given statement claims that for any function f that is increasing on an interval [a, b], the minimum value of f(x) on that interval is reached at x = a i.e., f(a).

Step 2: Analyze the statement

It helps to note that the function f is assumed to be defined and continuous on the closed interval [a, b]. If this is the case, then by definition of an increasing function, for any two numbers x and y in [a, b], if x < y, then f(x) < f(y). Thus, considering the endpoints of the interval, at x = a, we will have f(a) ≤ f(x) for any x in [a, b]. This means that the minimum value of f(x) is indeed at x = a.

Step 3: Provide the final answer

Considering the definition of an increasing function, and assuming that the function is continuous and defined on the interval [a, b], the statement is True.

Key Concepts

Minimum Value

When we talk about the minimum value of a function on a given interval, we're referring to the smallest output the function can produce on that interval. For a function that is increasing on a closed interval

• The value at the starting point of the interval, often expressed as \( f(a) \) where the interval is \([a, b]\), is the smallest.
• In simpler terms, as you move from left to right along an increasing function, the values on the graph keep growing.
• Therefore, the function's lowest point in the interval is right where you begin, at \( x = a \).

For example, think of a situation where you're walking uphill from point \(a\) to point \(b\). The lowest point is at \(a\), marking the start of your climb. As you continue upward towards \(b\), your elevation increases. Consequently, the minimum point is where the hike begins, at \( x = a \).
Understanding this concept helps in tackling problems that require assessing the lowest output of increasing functions on closed intervals.

Continuous Function

Continuous functions play a crucial role in mathematics because they don't have any breaks, jumps, or holes on their graphs. This smoothness implies that for every point between the endpoints of an interval, there’s a corresponding point on the graph.

• Because continuous functions are well-behaved without sudden interruptions, they are predictable and easier to work with.
• Analyzing such functions is straightforward because changes are uniform; you won’t suddenly find a missing point or a sharp turn in the graph.

In the context of increasing functions, continuity means that as \( x \) moves from \(a\) to \(b\), \( f(x) \) keeps maintaining its growth smoothly. If a function is continuous over a closed interval, we can confidently expect it to gradually shift from lower values to higher values without any surprise dips or spikes.

Closed Interval

The term "closed interval" refers to an interval that includes both of its endpoints. We symbolize closed intervals with square brackets, such as \([a, b]\). This inclusion means that the interval contains every point from \(a\) to \(b\), plus the endpoints themselves. It's important because:

• In a closed interval, a function is considered over the entire span from \(x = a\) to \(x = b\), accounting for the end values.
• This inclusivity ensures no missing pieces when determining properties like minimum and maximum values.

In problems involving increasing functions, the closed interval is vital. The function’s behavior is evaluated precisely from \(a\) to \(b\). Knowing that \(f(a)\) and \(f(b)\) are part of our analysis is critical, ensuring completeness of our evaluation over the entire interval.