Problem 29 - Single Variable Calculus

Question

Suppose that the equation $$ a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x=0 $$ has a positive root \(r\). Show that the equation $$ n a_{n} x^{n-1}+(n-1) a_{n-1} x^{n-2}+\cdots+a_{1}=0 $$ has a positive root smaller than \(r\). Hint: Use Rolle's Theorem.

Step-by-Step Solution

Answer

In summary, given a polynomial equation \[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x = 0 \] with a positive root \(r\), the derivative of the polynomial is \[ f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1 \]. By applying Rolle's Theorem on the interval (0, r), we have shown that there exists a positive root c, with \(0 < c < r\), such that \( f'(c) = 0 \). Therefore, the derivative equation has a positive root smaller than r.

Step 1: Identify the given polynomial and its positive root

The given polynomial equation is: \[ a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x = 0 \] and it has a positive root r.

Step 2: Find the derivative of the polynomial

By using the power rule for derivatives, we will find the derivative of the given polynomial equation. The derivative is: \[ f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1 \]

Step 3: Apply Rolle's Theorem

According to Rolle's Theorem, there must exist a number c between 0 and r such that the derivative is 0. Let's see how the conditions of Rolle's Theorem apply: - f(x) is continuous on [0, r], as a polynomial is continuous everywhere - f(x) is differentiable on (0, r), as a polynomial is differentiable everywhere - f(0) = 0: Simply substitute x = 0 and see \[ f(0) = a_n \cdot 0^n + a_{n-1} \cdot 0^{n-1} + \cdots + a_1 \cdot 0 = 0 \] - f(r) = 0: by definition, as r is a root of the polynomial Given that all the conditions for Rolle's Theorem are met, there exists a number c in the open interval (0, r) such that f'(c) = 0.

Step 4: Conclude that the derivative has a positive root smaller than r

Since f'(c) = 0, c is a root of the derivative, and since c is in the open interval (0, r), this positive root c is smaller than r. Thus, the equation \( n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \cdots + a_1 = 0 \) has a positive root smaller than r.

Key Concepts

Polynomials

In mathematics, polynomials are algebraic expressions consisting of variables and coefficients combined using only addition, subtraction, multiplication, and non-negative integer exponents. The general form of a polynomial is given by: \[ a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_1 x + a_0 \]where \(a_n\), \(a_{n-1}\), ..., \(a_0\) are constants called coefficients, and the variable \(x\) represents the unknown. Polynomials are fundamental to many areas of mathematics, including calculus and algebra.

• Polynomials can be of various degrees, determined by the highest power of the variable. For instance, the polynomial \(x^3 + 3x^2 + 3x + 1\) is a cubic polynomial.
• They are continuous and smooth functions, meaning they have no breaks, jumps, or corners in their graphs.
• Polynomials are used to model a range of real-world scenarios where the relationship between variables is proportional or consistent over intervals.

Understanding how to work with polynomials is crucial in solving equations where these expressions are set equal to zero.

Derivatives

Derivatives represent the rate of change of a function with respect to a variable. In simple terms, a derivative shows how a function changes as its input changes. The process of finding the derivative of a function is known as differentiation.To find the derivative of a polynomial function, each term is differentiated separately using the power rule. The power rule states that for \(x^n\), the derivative is \(n \cdot x^{n-1}\).

• For example, given the polynomial \(f(x) = a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x\), the derivative is \(f'(x) = n a_n x^{n-1} + (n-1) a_{n-1} x^{n-2} + \text{...} + a_1\).
• Derivatives help determine critical points of a function, such as maxima, minima, and points of inflection.
• In applied mathematics, derivatives are used to calculate speeds, accelerations, and in general any changing quantity.

Mastering derivatives is key to understanding the behavior of polynomial graphs and their rate of change.

Roots of Equations

The roots of an equation are values for which the equation holds true, meaning that when these values are substituted into the equation, the result is zero. In the case of polynomial equations, roots are crucial in understanding the solutions to the equations and are often referred to as "solutions" or "zeros."For a polynomial \(a_n x^n + a_{n-1} x^{n-1} + \text{...} + a_1 x + a_0 = 0\), the values of \(x\) that satisfy the equation are its roots.

• Roots are where the graph of the polynomial intersects the x-axis.
• Polynomials of degree \(n\) can have up to \(n\) roots, though some might be complex or repeated.
• Finding roots can be achieved through various methods such as factoring, synthetic division, or using the quadratic formula for second-degree polynomials.

In the context of this problem, finding roots of the derivative of a polynomial helps locate points where the original polynomial has particular properties like maxima and minima.

Continuity and Differentiability

Continuity and differentiability are important properties of functions that tell us how smooth the graph of a function is. A function is continuous if there is no interruption in its graph—essentially, you can draw the function without lifting your pen.

• All polynomials are continuous functions over all real numbers. This means they do not have any jumps, holes, or vertical asymptotes.

Differentiability refers to whether a function has a derivative at each point in its domain. A function is differentiable at a point if its graph does not have any sharp corners or cusps at that point.

• Polynomials are differentiable everywhere, as they are composed of smooth curves.

In the exercise, these properties ensure that Rolle's Theorem can be applied. Rolle's Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, and if the function has the same value at two points on the interval, then there is at least one point between them where the function's derivative is zero. This is key in proving the existence of roots within certain intervals.