Problem 13 - Calculus: Early Transcendentals

Question

Let \(f\) be differentiable on \(\mathbb{R}\). Suppose that \(f^{\prime}(x) \neq 0\) for every \(x\). Prove that \(f\) has at most one real root.

Step-by-Step Solution

Answer

The function \(f\) can have at most one real root because assuming otherwise leads to a contradiction with Rolle's Theorem.

Step 1: Understand the Problem

We are given that a function \(f\) is differentiable on \(\mathbb{R}\) and its derivative \(f'(x)\) is never zero for any real \(x\). We need to prove that this function \(f\) can have at most one real root, meaning at most one value \(x = a\) such that \(f(a) = 0\).

Step 2: Assume Multiple Roots Exist

Suppose, for contradiction, that \(f\) has two distinct real roots, say \(a\) and \(b\), such that \(f(a) = 0\) and \(f(b) = 0\) where \(a eq b\).

Step 3: Apply Rolle's Theorem

According to Rolle's Theorem, if a function is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \(f(a) = f(b) = 0\), then there exists some \(c\) in \((a, b)\) such that \(f'(c) = 0\).

Step 4: Contradiction Arises

Since we assumed \(f'(x) eq 0\) for all real \(x\), it's impossible for there to exist such a \(c\) where \(f'(c) = 0\). This contradicts our earlier assumption that there are two distinct roots \(a\) and \(b\).

Step 5: Conclusion

Since the assumption of having two distinct roots leads to a contradiction, \(f\) can have at most one real root.

Key Concepts

Real Roots

A real root of a function is a solution to the equation \( f(x) = 0 \). Finding these roots can be critical because they tell us where the function crosses the x-axis. For a function \( f \) that is differentiable across all real numbers, if we identify a real number \( a \) such that \( f(a) = 0 \), then \( a \) is the real root of that function.
Understanding real roots helps us gain insights into the behavior of function analytics. This can include:

• Determining intercepts: Points where the graph meets the x-axis.
• Analyzing changes in direction: Real roots can signal points where a graph changes its course.
• Solving equations: They help in finding solutions to polynomial and other complex equations.

However, the unique aspect in this problem is that the function \( f \) can have at most one real root, given the condition that \( f'(x) eq 0 \), which plays a significant role in limiting the possibilities of multiple real roots.

Rolle's Theorem

Rolle's Theorem is a central concept in calculus that applies to functions on continuous intervals. It asserts that if \( f \) is continuous on a closed interval \([a, b]\), differentiable on an open interval \((a, b)\), and \( f(a) = f(b) \), then there must be at least one \( c \) in the interval \((a, b)\) where the derivative is zero, i.e., \( f'(c) = 0 \).
This theorem is particularly useful because it provides a formal way to prove the presence of stationary points under specific conditions:

• It ensures that between any two identical function values, there is a point where the slope is zero.
• This must inevitably occur when the endpoints of an interval are of equal value.
• Knowing \( f'(c) = 0 \) can be pivotal in identifying potential maximum or minimum points.

In the context of the given problem, applying Rolle's Theorem under the assumption of multiple roots reveals a contradiction, since \( f'(x) \) cannot be zero at any point, thus proving that more than one root cannot exist.

Contradiction in Proofs

Contradiction is a common logical technique used in mathematical proofs. It involves assuming that a statement is true and then showing that this assumption leads to a logical inconsistency. In this way, you demonstrate that the assumption must be false, thus proving the statement.
Here’s how a proof by contradiction typically works:

• Assume the opposite of what you want to prove.
• Use logical reasoning and established theorems to find a contradiction.
• Once the contradiction is found, conclude that the initial assumption is false, confirming the original statement.

In our exercise, we assume that \( f \) has two real roots. When applying Rolle's Theorem, this assumption leads to a requirement that \( f'(c) = 0 \), which is contradictory because we are given that \( f'(x) eq 0 \) for all \( x \). This contradiction highlights that the function can only have one or no real root. Contradiction in proofs is a powerful method to establish valid conclusions when direct methods are complex or impractical.