Problem 64 - Calculus Single Variable

Question

Show that \(x^{3}-3 x^{2}+4 x-1=0\) has exactly one real root.

Step-by-Step Solution

Answer

The polynomial is strictly increasing and crosses the x-axis exactly once, confirming one real root.

Step 1: Analyze the polynomial to apply the derivative

Identify the function as the polynomial: \[ f(x)=x^{3}-3x^{2}+4x-1. \] Next, take the derivative \( f'(x) \) to examine its behavior: \[ f'(x)=3x^{2}-6x+4. \]

Step 2: Find critical points of the derivative

Set the derivative equal to zero to find potential points where the function may change direction:\[ 3x^{2}-6x+4=0. \]Solve using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), where \( a=3, b=-6, \) and \( c=4 \). Compute: \[ b^{2} - 4ac = (-6)^{2} - 4 \times 3 \times 4 = 36 - 48 = -12. \] Since the discriminant is negative, there are no real roots for the derivative. This means \( f(x) \) does not have any critical points in real numbers.

Step 3: Analyze the behavior and monotonicity of the original polynomial

Since the derivative \( f'(x) = 3x^{2}-6x+4 \) has no real roots, it is never zero, implying it does not change sign. Evaluate \( f'(x) \) at a point, such as \( x = 0 \):\[ f'(0) = 3(0)^{2} - 6(0) + 4 = 4. \] Since \( f'(0) > 0 \) and it doesn't change sign, \( f(x) \) is strictly increasing across all real numbers.

Step 4: Confirm continuity and end behavior of the polynomial

Polynomials are continuous and differentiable over the entire real line. Evaluate end behavior: as \( x \to -\infty \) or \( x \to +\infty \), the leading term \( x^{3} \) dominates:- As \( x \to +\infty \), \( x^{3} \to +\infty \), hence \( f(x) \to +\infty \).- As \( x \to -\infty \), \( x^{3} \to -\infty \), hence \( f(x) \to -\infty \).By the Intermediate Value Theorem, \( f(x) \), being continuous and increasing, crosses the x-axis exactly once.

Key Concepts

Polynomial function

A polynomial function is a mathematical expression that involves variables raised to whole number powers and has coefficients. These functions are constructed through the operations of addition, subtraction, multiplication, and in rare cases, division by constants, but never division by a variable.
Consider the polynomial function given in the exercise:

• \( f(x) = x^{3} - 3x^{2} + 4x - 1 \) is a third-degree polynomial because the highest power of the variable \( x \) is 3.

Such functions can be represented generally as \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where \( n \) is a non-negative integer and the coefficients \( a_n, a_{n-1}, \ldots, a_0 \) are real numbers. The behavior of polynomial functions, like their end behavior or the number of real roots, can be predicted by examining their structure closely. This forms the basis for further operations like differentiation.

Derivative

The derivative of a function describes how the function's output changes with respect to a change in the input. This allows us to find the slope of the tangent line to the function at any given point. Understanding the derivative is crucial for analyzing the behavior of functions, such as identifying when they are increasing or decreasing.
For the polynomial function

• \( f(x) = x^3 - 3x^2 + 4x - 1, \)

the first derivative is:

• \( f'(x) = 3x^2 - 6x + 4. \)

Taking the derivative helps determine critical points, which occur where \( f'(x) = 0 \) or where the function is not differentiable. However, for our polynomial, \( f'(x) \) does not equal zero for any real \( x \) due to a negative discriminant (- no real solution exists). This lack of real critical points indicates the function must be strictly increasing or decreasing across its entire domain. By checking the sign of the derivative, we can confirm the increasing or decreasing nature of the function.

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is an essential concept for understanding the behavior of continuous functions. It states that if a function \( f(x) \) is continuous on a closed interval \( [a, b] \), and \( N \) is any number between \( f(a) \) and \( f(b) \), there exists at least one \( c \) within \( (a, b) \) such that \( f(c) = N \).
Given the exercise, we see that \( f(x) \) (a polynomial) is continuous over the entire real number line. Observing the end behavior:

• As \( x o +\infty, f(x) o +\infty \)
• As \( x o -\infty, f(x) o -\infty \)

This implies that the function covers all values from negative to positive infinity. By IVT, since the function transitions from negative to positive as \( x \) varies from \( -\infty o +\infty \), it must cross the x-axis at least once. Coupled with the derivative analysis showing monotonicity, it crosses exactly once, confirming exactly one real root.

Monotonicity

Monotonicity refers to the property of a function to be consistently increasing or decreasing over its domain. Analyzing the monotonic behavior of a function helps in predicting how the function behaves across its domain. A strictly monotonic function never changes direction, meaning it only either increases or decreases.
For the polynomial function in question:

• The derivative \( f'(x) = 3x^2 - 6x + 4 \) is never zero due to the negative discriminant.
• Calculating \( f'(0) = 4 \) (a positive value) indicates that \( f(x) \) is strictly increasing as the derivative is positive.

This consistent increase establishes that the function has no local maxima or minima within the real number line. In this context, monotonicity aids in determining the number of real roots, as an increasing function can intersect the x-axis only once, forecasting a single real solution as concluded in the original exercise solution.