Question

If $$ C_{0}+\frac{C_{1}}{2}+\cdots+\frac{C_{n-1}}{n}+\frac{C_{n}}{n+1}=0 $$ where \(C_{0}, \ldots, C_{n}\) are real constants, prove that the equation $$ C_{0}+C_{1} x+\cdots+C_{n-1} x^{\prime-1}+C_{n} x^{n}=0 $$ has at least one real root between 0 and 1 .

Short answer

Question: Prove that if \(C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \cdots + \frac{C_{n-1}}{n} + \frac{C_n}{n+1} = 0\), then the polynomial equation \(C_0 + C_1 x + \cdots + C_{n-1} x^{n-1} + C_n x^n = 0\) has at least one real root between 0 and 1.

Step-by-step solution

Define a function f(x)

To make the problem easier to work with, let's define a polynomial function \(f(x)\) with terms involving the constants \(C_i\) as $$ f(x) = C_0x + C_1\frac{x^2}{2} + C_2\frac{x^3}{3} + \cdots + C_{n-1}\frac{x^n}{n} + C_n\frac{x^{n+1}}{n + 1}. $$

Determine values of f(x) at x=0 and x=1

First, let us evaluate the function \(f(x)\) at \(x=0\) and \(x=1\): $$ f(0) = C_0 \cdot 0 + C_1 \cdot 0 + \cdots + C_{n-1} \cdot 0 + C_n \cdot 0 = 0 $$ and $$ f(1) = C_0 + \frac{C_1}{2} + \frac{C_2}{3} + \cdots + \frac{C_{n-1}}{n} + \frac{C_n}{n+1}. $$ Notice that from the given condition of the exercise, \(f(1) = 0\) as well.

Check the requirements of Rolle's theorem

Clearly, \(f(x)\) is a polynomial function, which means it is continuous and differentiable on the interval \((0, 1)\). Furthermore, we established that \(f(0) = f(1) = 0\). These conditions satisfy the requirements for Rolle's theorem.

Apply Rolle's theorem and find f'(x)

Since \(f(x)\) meets the conditions of Rolle's theorem, there must exist some \(c \in (0, 1)\) such that \(f'(c) = 0\). Let's first find \(f'(x)\): $$ f'(x) = C_0 + C_1 x + \cdots + C_{n-1}x^{n-1} + C_n x^n. $$

Conclusion

By Rolle's theorem, there exists a \(c \in (0, 1)\) such that \(f'(c) = 0\). Since \(f'(x)\) is the same as the given polynomial equation, we can conclude that there exists at least one real root, \(c\), between 0 and 1 for the equation $$ C_0 + C_1 x + \cdots + C_{n-1} x^{n-1} + C_n x^n = 0. $$

Key concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of terms, each consisting of a constant coefficient multiplied by a variable raised to a non-negative integer power. In simpler terms, they are functions of the form \(a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0\), where the coefficients \(a_i\) are real numbers and \(x\) is the variable. These functions are quite versatile and can represent various types of mathematical models, from simple linear equations to more complex curves.

Here are some key characteristics of polynomial functions:

• They are defined for all real numbers, meaning you can input any real number into the function;
• Polynomials are smooth and continuous, which means they do not have any breaks or holes in their graph;
• The degree of a polynomial, which is the highest power of \(x\), determines the general shape of its graph and the maximum number of turning points the graph can have.

In the context of the exercise, the polynomial function \(f(x)\) is defined with terms involving the constants \(C_i\), each term being a polynomial itself. Understanding these basic properties of polynomial functions helps us analyze the behavior and roots of the given equation.

Real Roots

The concept of real roots refers to the solutions of an equation where the polynomial equals zero. In simpler terms, a real root is a value for the variable that makes the equation true when substituted into the polynomial. Real roots are important because they show where the graph of the function intersects the x-axis.

Why are real roots significant?

• Finding real roots helps in understanding the behavior of polynomials, such as determining where the function changes signs;
• They have practical applications in various fields such as engineering, physics, and economics;
• Real roots can represent critical points or solutions to optimization problems.

In our exercise, we are concerned with proving that the equation \(C_0 + C_1 x + \cdots + C_{n-1} x^{n-1} + C_n x^n = 0\) has at least one real root between 0 and 1. This involves checking the necessary conditions, which include continuity and differentiability, as stipulated in Rolle's Theorem. The existence of such a root means that there is a point \(c\) where the polynomial equals zero.

Derivatives

Derivatives are a fundamental concept in calculus representing the rate of change of a function with respect to its variable. They provide a way to determine the slope of the tangent line to the graph of the function at any given point. In essence, the derivative \(f'(x)\) of a function \(f(x)\) describes how \(f(x)\) changes as \(x\) changes.

Here’s why derivatives are useful:

• They help in finding the maximum and minimum points of functions, which are critical for optimization problems;
• Derivatives can be used to identify points of inflection, where the curvature of a graph changes;
• They allow us to apply the Mean Value Theorem and Rolle's Theorem, which are crucial in proving the existence of roots.

In the given problem, once the function \(f(x)\) is defined, its derivative \(f'(x)\) is calculated. This derivative represents the polynomial equation we are concerned with. By Rolle's Theorem, since \(f(0) = f(1) = 0\) and \(f(x)\) is continuous and differentiable, there must exist some \(c\) between 0 and 1 such that \(f'(c) = 0\). This is pivotal in proving there exists at least one real root \(c\) for the polynomial equation within the interval \(0, 1\).