Problem 58 - Thomas Calculus

Question

Rolle's Theorem \begin{equation} \begin{array}{l}{\text { a. Construct a polynomial } f(x) \text { that has zeros at } x=-2,-1,0} \\ \quad {\text { 1, and } 2 .} \\ {\text { b. Graph } f \text { and its derivative } f^{\prime} \text { together. How is what you see }} \\ \quad {\text { related to Rolle's Theorem? }} \\ {\text { c. Do } g(x)=\sin x \text { and its derivative } g^{\prime} \text { illustrate the same }} \\ \quad {\text { phenomenon as } f \text { and } f^{\prime} ?}\end{array} \end{equation}

Step-by-Step Solution

Answer

The polynomial \( f(x) \) and \( g(x) = \sin x \) both illustrate Rolle's Theorem, showing stationary points between zeros.

Step 1: Define the Polynomial

Rolle's Theorem requires that the function be continuous on the closed interval and differentiable on the open interval. To construct a polynomial function with zeros at \( x = -2, -1, 0, 1, \) and \( 2 \), we can use the fact that these zeros correspond to the roots of the polynomial. We can write the function as \( f(x) = (x+2)(x+1)x(x-1)(x-2) \). This polynomial is continuous and differentiable everywhere, satisfying the conditions for Rolle's Theorem.

Step 2: Graph the Function and Its Derivative

Plot the graph of \( f(x) = (x+2)(x+1)x(x-1)(x-2) \) and its derivative. The derivative \( f'(x) \) can be obtained by using the product rule. The graph of \( f(x) \) should show where the function intersects the x-axis. Between every pair of adjacent roots, \( f'(x) \) should indicate where it crosses the x-axis, confirming that \( f(x) \) has stationary points.

Step 3: Apply Rolle's Theorem

Rolle's Theorem states that if a function \( f \) is continuous on \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) \), there is at least one \( c \) in the interval \((a, b)\) such that \( f'(c) = 0 \). On the graph, at each interval between the zeros \(-2, -1\), \(-1, 0\), \(0, 1\), and \(1, 2\), there must be at least one point where the slope of the tangent to the curve is zero, which is evidenced by the derivative crossing the x-axis.

Step 4: Analyze Sine Function

Consider the function \( g(x) = \sin x \) and its derivative \( g'(x) = \cos x \). Although \( \sin x \) is not a polynomial, within each interval where \( g(x) \) crosses the x-axis (like \([0, \pi]\), \([\pi, 2\pi]\), etc.), \( g(x) \) returns to the same value, showing derivatives crossing the x-axis at \( \pi/2, 3\pi/2, \ldots \), similar to the polynomial case.

Step 5: Evaluate Rolle's Theorem Applicability

Both the polynomial \( f(x) \) and the trigonometric function \( g(x) = \sin x \) satisfy the same phenomenon predicted by Rolle's Theorem: the existence of a point where the derivative is zero between any two zeros of the function. This confirms that the properties described by Rolle's Theorem are not exclusive to polynomial functions but apply to any function that meets the criteria.

Key Concepts

Polynomial Functions

Polynomial functions are essential in mathematics due to their simple yet flexible structure. A polynomial function can be represented as a sum of terms, each consisting of a variable raised to a non-negative integer power. The general form is:\[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \ + a_1x + a_0 \]where the coefficients \( a_n, a_{n-1}, \ldots, a_0 \) are real numbers, and \( n \) is the degree of the polynomial (the highest power of \( x \)).
Polynomials are classified based on their degree:

• Linear polynomial: Degree 1 (e.g., \( x + 2 \))
• Quadratic polynomial: Degree 2 (e.g., \( x^2 - 4x + 4 \))
• Cubic polynomial: Degree 3 (e.g., \( x^3 + 3x^2 + 3x + 1 \))

In the exercise, the polynomial \( f(x) = (x+2)(x+1)x(x-1)(x-2) \) is a quintic polynomial (degree 5) with zeros at \( x = -2, -1, 0, 1, \) and \( 2 \). Polynomials are smooth and continuous, making them suitable for applying Rolle's Theorem.

Derivatives

The derivative of a function provides crucial information about its behavior. It represents the rate of change or the slope of the function at any given point. Mathematically, it is denoted by \( f'(x) \) for the function \( f(x) \).
To find the derivative of a polynomial, the power rule is often used, where the derivative of \( x^n \) is \( nx^{n-1} \). However, in more complex functions, rules like the product rule could be necessary. The product rule states:\[ (uv)' = u'v + uv' \]where \( u \) and \( v \) are functions of \( x \).
For the polynomial \( f(x) = (x+2)(x+1)x(x-1)(x-2) \), applying the product rule several times or expanding the polynomial before differentiating might be required. Once the derivative \( f'(x) \) is determined, it can be used to identify local maxima, minima, and points where the slope is zero, connecting back to Rolle's Theorem.

Zeros of a Function

Zeros of a function, also known as roots, are the values of \( x \) for which the function \( f(x) \) equals zero. Analyzing the zeros of a polynomial plays a key role in understanding its graph and the solutions to related equations.
In the polynomial \( f(x) = (x+2)(x+1)x(x-1)(x-2) \), the factors

• \( (x+2) \)
• \( (x+1) \)
• \( x \)
• \( (x-1) \)
• \( (x-2) \)

indicate the zeros are at \( x = -2, -1, 0, 1, \) and \( 2 \). Understanding the location of zeros helps visualize where the polynomial graph will intersect the x-axis. Additionally, between each adjacent pair of zeros, Rolle's Theorem assures at least one point where the derivative (or slope) of the polynomial is zero.

Graphing Functions

Graphing functions provides a visual interpretation, helping to understand their behavior, especially in terms of continuity and differentiability. For polynomial functions, the graph is a smooth curve, unaffected by breaks or cusps.
To graph a function, key elements include:

• Identifying zeros where the function crosses the x-axis
• Using derivatives to locate peaks, troughs, and points of inflection
• Plotting the function's derivative to observe changes in slope

In the context of Rolle's Theorem, plotting both \( f(x) \) and its derivative \( f'(x) \) is crucial. The exercise reveals that between every pair of zeros of \( f(x) \), there must be at least one zero of \( f'(x) \), illustrating stationary points. These points occur where the tangent to the graph is horizontal, showcasing the theorem's real-world applicability.