Problem 24 - College Algebra

Question

For the following exercises, use the Intermediate Value Theorem to confirm that the given polynomial has at least one zero within the given interval. $$ f(x)=x^{3}-9 x, \text { between } x=-4 \text { and } x=-2 $$

Step-by-Step Solution

Answer

The function has a zero between \(-4\) and \(-2\) because \(f(-4)\) and \(f(-2)\) have opposite signs.

Step 1: Understand the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\), and \( f(a) \) and \( f(b) \) have opposite signs, then there exists at least one \( c \) in the interval \((a, b)\) such that \( f(c) = 0 \). This means the function crosses the x-axis at least once in the interval.

Step 2: Define the Function and Check Continuity

The function given is \( f(x) = x^3 - 9x \). This is a polynomial function, and polynomial functions are continuous everywhere on the real axis, including the interval \([-4, -2]\). Thus, \( f(x) \) is continuous on \([-4, -2]\).

Step 3: Evaluate the Function at the Endpoints

Calculate \( f(-4) \) and \( f(-2) \) to check their signs:\[ f(-4) = (-4)^3 - 9(-4) = -64 + 36 = -28 \]\[ f(-2) = (-2)^3 - 9(-2) = -8 + 18 = 10 \]

Step 4: Determine the Sign Change

\( f(-4) = -28 \), which is negative, and \( f(-2) = 10 \), which is positive. Since \( f(-4) \) and \( f(-2) \) have opposite signs, the Intermediate Value Theorem confirms there is at least one zero of \( f(x) \) in the interval \((-4, -2)\).

Key Concepts

Polynomial Function

A polynomial function is a mathematical expression consisting of variables raised to whole number powers and coefficients. In simpler terms, it's an expression like

• \( f(x) = x^3 - 9x \)

This function has variables raised to powers, specifically the cube and linear terms in this example. Polynomials are built from terms added or subtracted together, like \( x^3 \) and \(-9x \). They're found in patterns and structures across mathematics. Notably, polynomial functions are very smooth. There's no sudden jumps or breaks, which brings us to their smooth nature, known as continuity.

Continuity of Functions

Continuity essentially means a function is uninterrupted; it flows without any jumps or gaps. This is easily visualized like drawing a line on a piece of paper without lifting your pencil. A function is continuous over an interval if, within that interval, you can move from one point to another without a break in the graph. For polynomial functions, like the one mentioned earlier \( f(x) = x^3 - 9x \), continuity is guaranteed everywhere on the real number line.

• No jumps: No sudden leaps in values.
• No breaks: No gaps in its domain.
• Predictable behavior: Easily understood and calculated.

Thus, when we're working in any interval, such as \([-4, -2]\), a polynomial remains continuous there, assuring us it covers all intermediate values between its endpoints.

Intermediate Value Theorem Application

The Intermediate Value Theorem is a fundamental concept used to demonstrate the presence of roots (or zeros) within a specified interval. Its premise is simple yet powerful: if you have a continuous function over a closed interval and the function takes on different signs at the endpoints of that interval, then the function must cross zero somewhere within it.For the function \( f(x) = x^3 - 9x \), consider the interval \([-4, -2]\):

• Calculate \( f(-4) \): This results in \(-28\), a negative number.
• Calculate \( f(-2) \): This yields \(10\), a positive number.

Since these values have opposite signs, the Intermediate Value Theorem guarantees that somewhere in \((-4, -2)\), the function crosses the x-axis. Specifically, there is at least one value \( c \) where \( f(c) = 0 \). This is crucial for finding solutions to many real-world and theoretical problems where determining the point of change is necessary.