Problem 97 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Suppose \(f\) is continuous on \([a, b]\) and \(f(a)

Step-by-Step Solution

Answer

This situation does not contradict the Intermediate Value Theorem. The theorem states that for a continuous function \(f\) on the interval [a, b], there exists a number \(a < c < b\) such that \(f(c) = M\) only if \(M\) is between \(f(a)\) and \(f(b)\). Since the given value of \(M\) lies outside the interval [\(f(a)\), \(f(b)\)], the conditions of the theorem are not satisfied, and hence there is no contradiction.

Step 1: Recall the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function f is continuous on a closed interval [a, b], then for any value M between f(a) and f(b), there exists a number a < c < b such that f(c) = M.

Step 2: Analyze the given information

We are given: 1. f is continuous on [a, b]. 2. f(a) < f(b). 3. M is a number outside the interval [f(a), f(b)].

Step 3: Check if there exists a number a < c < b such that f(c) = M

Since M is outside the interval [f(a), f(b)], it means that either M < f(a) or M > f(b). In either case, M is not between the values f(a) and f(b).

Step 4: Reason if this contradicts the Intermediate Value Theorem

The Intermediate Value Theorem states that there exists a number a < c < b such that f(c) = M, only if M is between f(a) and f(b). In this case, M is not between f(a) and f(b), so the theorem doesn't guarantee the existence of such a number c. Thus, the situation does not contradict the Intermediate Value Theorem because the theorem's conditions are not met for this particular value of M.

Key Concepts

Continuous Function

A continuous function is a foundational concept in calculus and analysis, providing a mathematical model for a seamless, unbroken curve. Such a function, when graphed, will have no gaps, jumps, or sudden changes in direction. Essentially, you can draw it without lifting your pencil from the paper.

In more technical terms, a function is continuous at a point if the limit of the function as it approaches the point from either direction equals the value of the function at that point. In an even simpler sense, if you can trace the function smoothly at every point within its domain, it's continuous there.

To be formally continuous on a closed interval, such as \[a, b\], the function must meet the following criteria:

• The function is continuous at every point in the interval (no gaps or jumps).
• The limits from the left and the right at the endpoints exist and equal the function's actual values at those points.

This smoothness allows us to apply powerful theorems, like the Intermediate Value Theorem, to make predictions about the function's behavior within that interval.

Closed Interval

Understanding the significance of a closed interval in mathematics is crucial for grasping many concepts in calculus. A closed interval, typically denoted as \[a, b\], is a range of numbers that includes all the values between a and b, including the endpoints themselves. That means 'a' and 'b' are part of the interval.

The notion of a closed interval is central when discussing continuity and the behavior of functions. When you're dealing with a closed interval, you are considering a complete set of values that are bounded and well-defined. In the context of continuous functions, closed intervals are essential since they define the domain over which properties like the Intermediate Value Theorem hold true.

Using closed intervals also ensures mathematical reasoning is applied to a finite and manageable set of possibilities. This helps in setting clear parameters for examining the characteristics of functions and predicting their values at given points within the interval.

Mathematical Reasoning

Mathematical reasoning is the logical process used in math to reach conclusions based on hypotheses or known facts. It includes deductive and inductive reasoning. Deductive reasoning is when we apply general rules to specific cases to derive conclusions, whereas inductive reasoning involves looking at particular instances and inferring a general rule.

In the context of the Intermediate Value Theorem, mathematical reasoning is used to infer the existence (or non-existence) of a particular value within a continuous function. The exercise in question uses deductive reasoning by starting with the established conditions of the theorem (a continuous function on a closed interval with changing function values), recognizing that a value outside of these changing values is not covered by the theorem, and thus concluding there is no contradiction when such a value does not correspond to any point within the interval.

Solid mathematical reasoning is what prevents us from falling into erroneous conclusions—such as mistaking the conditions of the theorem to mean that any value must have a corresponding point within the interval, which the theorem does not guarantee if the value is outside the range of the function on that interval.