Problem 250 - Precalculus

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. $$ f(x)=-2 x^{3}-x, \text { between } x=-1 \text { and } x=1 $$

Step-by-Step Solution

Answer

The function has a zero in the interval [-1, 1] by the Intermediate Value Theorem.

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 \( N \) is a number between \( f(a) \) and \( f(b) \), then there exists at least one \( c \) in the open interval \((a, b)\) such that \( f(c) = N \). To apply this theorem for zeroes, we need \( f(a) \) and \( f(b) \) to have opposite signs, which implies that \( N = 0 \) lies between them.

Step 2: Check Continuity of the Function

The given function is \( f(x) = -2x^3 - x \). Polynomial functions are continuous everywhere. Therefore, \( f(x) \) is continuous on the interval \([-1, 1]\).

Step 3: Calculate Function Values at the Interval Endpoints

Evaluate the function \( f(x) = -2x^3 - x \) at the endpoints of the interval. Calculate \( f(-1) \): \[f(-1) = -2(-1)^3 - (-1) = -2(-1) + 1 = 2 - 1 = 1\]Calculate \( f(1) \): \[f(1) = -2(1)^3 - 1 = -2(1) - 1 = -2 - 1 = -3\]

Step 4: Check for Opposite Signs

From Step 3, we found \( f(-1) = 1 \) and \( f(1) = -3 \). Since \( f(-1) \) is positive and \( f(1) \) is negative, the function values have opposite signs, confirming that there is at least one zero between \( x = -1 \) and \( x = 1 \) according to the Intermediate Value Theorem.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that consist of variables and coefficients, arranged in a sum of terms. Each term has a non-negative integer exponent. A simple example of a polynomial function is

• \( f(x) = 3x^2 + 2x + 1 \)

In our exercise, we have the polynomial function \( f(x) = -2x^3 - x \). This function is cubic due to the highest exponent being 3.
Polynomial functions are vital because they show a variety of behaviors and can model many real-world situations.
Each polynomial has a degree, which is determined by the highest power of the variable in the expression. Understanding polynomial functions is the foundation for grasping more complex concepts like continuity and the intermediate value theorem.

Continuity of Functions

The concept of continuity in functions means that the graph of the function is unbroken; you can draw it without lifting your pencil. In mathematical terms, a function is continuous if it does not have any jumps, breaks, or holes in its domain.
Polynomial functions are continuous everywhere. This continuity is critical when applying the Intermediate Value Theorem, as the theorem only holds for continuous functions.

• In our problem, the polynomial function \( f(x) = -2x^3 - x \) is continuous on the entire real number line, including the interval \([-1, 1]\).

Knowing a function's continuity helps in analyzing its behavior within a specified interval and is essential when looking for zeros within that range.

Zeros of Functions

Zeros of a function, also known as roots, are the values of \(x\) for which the function equals zero. For the function \(f(x)\), finding the zeros means solving the equation \(f(x) = 0\).
To identify a zero, we often use methods like factoring, utilizing the Intermediate Value Theorem, or numerical methods.
The Intermediate Value Theorem is particularly useful in confirming the existence of at least one zero in an interval when the function's values at the endpoints have opposite signs.

• In our exercise, we determined that \( f(-1) = 1 \) and \( f(1) = -3 \). Since one is positive and the other is negative, it confirms the presence of a zero between \(x = -1\) and \(x = 1\).

Finding zeros is crucial in many applications across different fields such as physics, engineering, and economics, where these roots often represent critical points or solutions.