Problem 72 - Calculus Early Transcendentals: Pearson New International Edition

Question

Suppose that \(f\) is continuous on \([a, b]\) and it is never zero there. Is it possible that \(f\) changes sign on \([a, b]\) ? Explain.

Step-by-Step Solution

Answer

No, it is not possible for such a function to change sign on \\([a, b]\\).

Step 1: Understanding Continuity on [a, b]

When a function is continuous on a closed interval \([a, b]\), it means that the function does not have any breaks, jumps, or points of discontinuity within that interval. The Intermediate Value Theorem plays a crucial role in understanding the behavior of continuous functions.

Step 2: Review of the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \(f\) is continuous on the interval \([a, b]\), and if \(N\) is any number between \(f(a)\) and \(f(b)\), then there is at least one \(c\) in \([a, b]\) such that \(f(c) = N\).

Step 3: Considering the Non-Zero Condition

The problem states that \(f(x)\) is never zero on \([a, b]\). For \(f\) to change sign from positive to negative or vice versa, it would need to cross zero. However, due to continuity and the non-zero condition, \(f(x)\) cannot equal zero at any point in \([a, b]\).

Step 4: Implication of Continuity and Non-Zero Condition

Since \(f(x)\) is both continuous and never zero on the interval \([a, b]\), this implies that \(f(x)\) must retain its sign across the entire interval. It cannot transition from positive to negative, or vice versa, without crossing zero, which the problem states is impossible.

Key Concepts

Continuity of Functions

Let's dive into the concept of continuity first. When we say a function is continuous on an interval, it means there are no gaps, jumps, or interruptions in its path over that interval. You can picture a continuous function as one you can draw without lifting your pencil from the paper.
For mathematicians, continuity is crucial because it ensures predictability in a function's behavior. The Intermediate Value Theorem relies heavily on this characteristic. If you're given that a function behaves smoothly over an interval, like \([a, b]\), whatever happens at the edge points (at \(a\) and \(b\)) must seamlessly transition through all points in between.
So, continuity tells us that if you choose any two points connected by a function in an interval, you can also find a connected path consisting of all intermediate points entirely mapped out by the function.

Closed Interval

The concept of a closed interval is vital when analyzing continuity. A closed interval, denoted as \([a, b]\), means that the endpoints \(a\) and \(b\) are included in the interval. This is different from an open interval, where endpoints are not part of the interval.
By including endpoints, you make sure the evaluation of a function at these points is considered. This is significant for many theorems and properties in calculus, including the Intermediate Value Theorem.
For the function \(f\) to be continuous in \([a, b]\), it must be defined and unbroken across the entire range, including at the beginning \(a\) and end \(b\).

• Closed intervals ensure no endpoint is neglected.
• They allow strict application and testing of continuity.

Sign Change in Functions

The concept of a function changing its sign is about shifting from positive to negative values, or vice versa. This involves crossing zero, making a critical examination necessary when a function is continuous on a closed interval and doesn't equal zero.
If you think about it, for a function to change sign, there must be a point where it touches zero. However, if a function is guaranteed to not be zero anywhere in a closed interval (like in our original problem), a sign change is impossible.
Given continuity, if our function starts positive and ends negative (or the opposite), the Intermediate Value Theorem ensures there should be a point crossing zero. Since that's not feasible here, our understanding concludes that the function retains a consistent sign throughout.

• Sign change requires a zero crossing.
• Continuity ensures the path doesn't suddenly jump over zero.