Problem 59 - Thomas Calculus

Question

Unique solution Assume that \(f\) is continuous on \([a, b]\) and differentiable on \((a, b) .\) Also assume that \(f(a)\) and \(f(b)\) have opposite signs and that \(f^{\prime} \neq 0\) between \(a\) and \(b .\) Show that \(f(x)=0\) exactly once between \(a\) and \(b .\)

Step-by-Step Solution

Answer

Given conditions ensure exactly one root in \((a, b)\).

Step 1: Understand the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists at least one number \( c \) in the interval \((a, b)\) such that \( f(c) = N \). Given that \( f(a) \) and \( f(b) \) have opposite signs, this means that 0 lies between \( f(a) \) and \( f(b) \), thus there exists some \( c \) where \( f(c) = 0 \).

Step 2: Use Bolzano's Theorem

Bolzano's Theorem is a special case of the Intermediate Value Theorem that concludes if a continuous function changes sign over an interval, there is a root in the interval. Since \( f(a) \) and \( f(b) \) have opposite signs, \( f(x) = 0 \) must have at least one root in \((a, b)\).

Step 3: Establish Uniqueness Using Differentiability

Since \( f \) is differentiable and \( f'(x) eq 0 \) everywhere in the interval \((a, b)\), the function does not have a horizontal tangent, and it cannot have multiple zero crossings within \( (a, b) \). If there were two distinct roots, by Rolle's Theorem, \( f'(x) = 0 \) at some point between the roots, contradicting our assumption that \( f'(x) eq 0 \) throughout \((a, b)\).

Step 4: Conclusion

Since \( f(x) \) is continuous and \( f(a) \) and \( f(b) \) are of opposite signs, there is at least one root within \((a, b)\). Further, the condition that \( f'(x) eq 0 \) ensures that there cannot be more than one root. Therefore, \( f(x) = 0 \) exactly once on the interval \((a, b)\).

Key Concepts

Bolzano's Theorem

Bolzano's Theorem is an essential concept in mathematics that helps us understand when a function has a root in a specific interval. This theorem is a more focused version of the Intermediate Value Theorem (IVT). Bolzano's Theorem specifically deals with continuous functions over a closed interval where the function values at the endpoints have opposite signs. Let's break it down:

• The function must be continuous on the interval [a, b]. This means there should be no breaks, jumps, or holes in the graph of the function within this interval.
• There should be a sign change between f(a) and f(b), meaning if one is positive, the other must be negative.

Due to these properties, Bolzano's Theorem guarantees that there is at least one point \( c \) in the interval (a, b) where \( f(c) = 0 \). This makes it extremely useful for determining the existence of roots. By ensuring continuity and different signs at endpoints, the theorem provides a logical certainty of a root's presence.

Rolle's Theorem

Rolle's Theorem is a fascinating result in calculus relating to differentiable functions. It gives conditions under which a function must have a horizontal tangent line between two points where the function values are equal:

• The function must be differentiable across the open interval (a, b).
• The endpoints of the interval must satisfy \( f(a) = f(b) \).
• The function should be continuous over the closed interval [a, b].

If these conditions hold, then there is at least one point \( c \) in the interval (a, b) such that the derivative \( f'(c) = 0 \). This means there is a point where the tangent is horizontal. In our scenario, since \( f'(x) eq 0 \), we use this theorem in reverse, confirming that if multiple roots existed, a zero derivative would occur, which is contradictory to our conditions.

Differentiability

Differentiability is a key concept in calculus that informs us about the behavior of a function. When we say a function is differentiable on an interval, we mean:

• A derivative exists at each point within that interval.
• The function is smooth and has no sharp corners or cusps in the interval.

A function being differentiable implies that it is also continuous within that interval, though not vice versa. In the context of our problem, differentiability of the function \( f \) ensures that the function does not have abrupt changes. It also allows us to apply tools like the derivative. Since the derivative \( f'(x) eq 0 \) is given, it suggests that the function does not flatten out or become horizontal, preventing multiple zero crossings in the interval.

Continuous Functions

Understanding continuous functions is crucial for solving problems involving theorems like Bolzano's and Rolle's. A function is continuous on an interval if:

• There are no breaks, jumps, or gaps in the function within that interval.
• For every value \( x \) in the interval, the value of the function approaches \( f(x) \) as closely as desired from both sides.

Continuity over a closed interval [a, b] implies that the function is predictable and behaves well without sudden disruptions. In many calculus problems, continuity is a prerequisite. Without it, the conclusions drawn from Bolzano's Theorem or Rolle's Theorem may fail. In our problem, continuous behavior between points a and b is crucial to assert the existence of the root \( f(x) = 0 \) and to use further differentiability properties to ensure it is unique.