Problem 33 - Calculus Early Transcendentals: Pearson New International Edition

Question

Prove: Let \(f\) be continuous on \([a, b]\) and differentiable on \((a, b)\). If \(f(a)\) and \(f(b)\) have opposite signs and if \(f^{\prime}(x) \neq 0\) for all \(x\) in \((a, b)\), then the equation \(f(x)=0\) has one and only one solution between \(a\) and \(b\). Hint: Use the Intermediate Value Theorem and Rolle's Theorem (Problem 22).

Step-by-Step Solution

Answer

The equation \(f(x) = 0\) has exactly one solution between \(a\) and \(b\).

Step 1: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem, since \(f\) is continuous on \([a, b]\) and \(f(a)\) and \(f(b)\) have opposite signs, there exists at least one \(c \in (a, b)\) such that \(f(c) = 0\). This ensures that there is a root within the interval \((a, b)\).

Step 2: Review the conditions of Rolle's Theorem

Rolle's Theorem states that if a function \(g\) is continuous on \([p, q]\), differentiable on \((p, q)\), and \(g(p) = g(q)\), then there exists some \(k \in (p, q)\) such that \(g'(k) = 0\).

Step 3: Apply the conditions of Rolle's Theorem to f(x)

Assuming there exist two roots, say \(c_1\) and \(c_2\) where \(f(c_1) = f(c_2) = 0\) and \(c_1 < c_2\), we can apply Rolle's Theorem to the interval \([c_1, c_2]\). According to Rolle's Theorem, there must be a point \(k\) in \((c_1, c_2)\) where \(f'(k) = 0\).

Step 4: Derive a contradiction

Since it is given that \(f'(x) eq 0\) for all \(x\) in \((a, b)\), the condition found from Rolle's Theorem (namely, that \(f'(k) = 0\)) leads to a contradiction. Thus, \(f(x) = 0\) cannot have two distinct roots \(c_1\) and \(c_2\), and must have exactly one solution.

Key Concepts

Rolle's Theorem

Rolle's Theorem is a fundamental theorem in calculus that helps us understand the behavior of differentiable functions. It tells us that under certain conditions, the derivative of the function, at a point within a given interval, must be zero.

When applying Rolle's Theorem, we need three conditions:

• The function must be continuous on the closed interval \[a, b\].
• The function must be differentiable on the open interval \(a, b\).
• The values of the function at the endpoints must be equal, i.e., \(f(a) = f(b)\).

If all these conditions are satisfied, there exists at least one point \(c\) within \(a < c < b\) where \(f'(c) = 0\).

In the context of the original problem, Rolle's Theorem is used to show that if a function had two roots within \(a, b\), then the derivative should be zero at least one point, contradicting the given condition.

Continuity

Continuity is a property of functions that ensures there are no breaks, jumps, or holes in its graph. A function \(f\) is continuous at a point \(x = c\) if the following conditions are satisfied:

• \(f(c)\) is defined.
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) is equal to \(f(c)\).

For a function to be continuous on an interval, it must be continuous at every point in that interval.

In our original exercise, continuity on the interval \[a, b\] is crucial because it allows us to use the Intermediate Value Theorem, ensuring that the function crosses the x-axis, and hence has a root, when the function values at \(a\) and \(b\) have opposite signs.

Differentiability

Differentiability is a characteristic of functions that allows us to find their derivatives. Essentially, a function is differentiable at a point if it has a well-defined tangent (not a vertical tangent) at that point.

If \(f\) is differentiable at \(x = c\), the derivative \(f'(c)\) exists, meaning the rate of change is not infinite. Differentiability implies continuity, but not all continuous functions are differentiable (considerable examples are functions with sharp corners like the absolute value function).

In the problem, it was specified that \(f\) is differentiable on the open interval \( (a, b) \), which ensures that the function behaves smoothly without any vertical tangents or corners. This condition is necessary to apply Rolle's Theorem, which helps in finding conditions about the occurrence of roots in a differentiable function.

Root of an Equation

The root of an equation is a solution that makes the equation equal zero. It represents an x-value where the graph of the function intersects the x-axis. In mathematical terms, if \(f(c) = 0\), then \(c\) is a root of the equation \(f(x) = 0\).

In the scenario of the problem we're considering, finding a root between \(a\) and \(b\) involves:

• Using the Intermediate Value Theorem to assert the existence of at least one root because \(f(a)\) and \(f(b)\) have opposite signs.
• Verifying that there's exactly one root by employing Rolle's Theorem and the fact that \( f'(x) eq 0 \).

This proof prevents the assumption of multiple roots by demonstrating that if two roots existed, it would contradict the conditions provided in the exercise by setting up impossible scenarios for the derivative.