Problem 25 - College Algebra with Corequisite Support

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. \(f(x)=x^{3}-9 x,\) between \(x=2\) and \(x=4 .\)

Step-by-Step Solution

Answer

The polynomial has at least one zero between \(x=2\) and \(x=4\).

Step 1: Understanding the Intermediate Value Theorem

The Intermediate Value Theorem states that if a function \( f \) is continuous on a closed interval \([a, b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists at least one number \( c \) in the interval \((a, b)\) such that \( f(c) = N \). In this exercise, we want to determine if there is a zero, meaning \( N = 0 \), within the interval \( [2, 4] \).

Step 2: Evaluating the Function at the Endpoints of the Interval

Calculate \( f(2) \) and \( f(4) \) for the polynomial function \( f(x) = x^3 - 9x \) to see if there is a change of sign, indicating a zero in the interval. \[ f(2) = 2^3 - 9 \times 2 = 8 - 18 = -10 \]\[ f(4) = 4^3 - 9 \times 4 = 64 - 36 = 28 \]

Step 3: Applying the Intermediate Value Theorem

Observe that \( f(2) = -10 \) and \( f(4) = 28 \). Since \( f(x) \) is continuous as a polynomial and there is a sign change from negative to positive within the interval \( [2, 4] \), by the Intermediate Value Theorem, there must be at least one zero of \( f(x) \) in the interval \( (2, 4) \).

Key Concepts

polynomial functions

Polynomial functions are expressions composed of variables and coefficients, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents of variables. They form an important class of mathematical functions. The general form of a polynomial function is expressed as: \[ f(x) = a_n x^n + a_{n-1} x^{n-1} + ext{...} + a_1 x + a_0 \] where:

• \(a_n, a_{n-1}, ext{...}, a_0\) are constant coefficients,
• \(n\) is a non-negative integer indicating the degree of the polynomial.

In the given exercise, \( f(x) = x^3 - 9x \) is a polynomial function of degree 3, as the highest power of \(x\) is 3. Polynomial functions are smooth and continuous over the real numbers, which makes them particularly useful when applying the Intermediate Value Theorem.

continuity

The concept of continuity is crucial in understanding polynomial functions and their behavior within an interval. A function is continuous on an interval if there are no breaks, jumps, or holes in its graph over that interval. In simpler terms, you can draw the graph of the function without lifting your pencil from the paper.
Polynomials, like \( f(x) = x^3 - 9x \), are naturally continuous functions. This means they do not have any abrupt changes in value over their domain. Continuity is important because the Intermediate Value Theorem (IVT) relies on a function being continuous over a specified interval.
In our example, because \( f(x) \) is continuous between \(x = 2\) and \(x = 4\), we can confidently apply the IVT to check for zeros within this interval.

zero of a function

The zero of a function, also known as a root, is the value of \(x\) that makes the function equal to zero. In mathematical terms, for a function \(f(x)\), a zero is a value \(c\) such that \( f(c) = 0 \). Finding zeros is important in many areas, like solving equations or determining critical points of functions.
For the polynomial \( f(x) = x^3 - 9x \), we are tasked to find at least one zero in the interval \([2, 4]\). To apply the Intermediate Value Theorem, we look for a sign change between \( f(2) \) and \( f(4) \). Since there is a transition from negative \( f(2) = -10 \) to positive \( f(4) = 28 \), the theorem guarantees the existence of at least one zero in \((2, 4)\). This reliable presence of a zero is what makes the Intermediate Value Theorem so helpful.

sign change

A sign change in the context of a function occurs when the output of the function shifts from negative to positive or vice versa, across an interval. This concept is critical when you are trying to determine the existence of zeros using the Intermediate Value Theorem.
To observe a sign change, evaluate the function at different points. If the function values have different signs, a sign change has occurred, indicating that the function crosses the x-axis or has a zero.
In our example, for \( f(x) = x^3 - 9x \), evaluating the endpoints we find \( f(2) = -10 \) and \( f(4) = 28 \). This indicates a sign change from negative to positive, hence confirming that a zero exists within the interval \([2, 4]\). This is a key step to apply the Intermediate Value Theorem effectively.

evaluating endpoints

When using the Intermediate Value Theorem, evaluating the function at the endpoints of an interval is a critical step. This helps to determine the possibility of a zero within that interval.
Start by calculating the function's values at the given endpoints. For our exercise with \( f(x) = x^3 - 9x \), the interval provided is from \(x = 2\) to \(x = 4\).

• At \(x = 2\): \( f(2) = 2^3 - 9 \, (2) = -10 \)
• At \(x = 4\): \( f(4) = 4^3 - 9 \, (4) = 28 \)

This evaluation shows a change from negative to positive which aligns perfectly with the conditions required by the Intermediate Value Theorem. Such results imply that the continuous polynomial function indeed has at least one zero between these endpoints.