Problem 246 - 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. $$ f(x)=x^{3}-9 x, \quad \text { between } x=-4 \text { and } x=-2 $$

Step-by-Step Solution

Answer

There is at least one zero for \(f(x) = x^3 - 9x\) between \(-4\) and \(-2\) because \(f(-4)\) and \(f(-2)\) have opposite signs.

Step 1: Understand the function

The given polynomial function is \( f(x) = x^3 - 9x \). We are tasked with using the Intermediate Value Theorem to confirm there is at least one zero between \(x = -4\) and \(x = -2\).

Step 2: Evaluate the function at the endpoints

Calculate the value of the function at the interval's endpoints. First, \( f(-4) \) is calculated as:\[f(-4) = (-4)^3 - 9(-4) = -64 + 36 = -28\]Next, calculate \( f(-2) \):\[f(-2) = (-2)^3 - 9(-2) = -8 + 18 = 10\]

Step 3: Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that for a continuous function, if \( f(a) \) and \( f(b) \) have different signs, there is at least one zero in the interval \((a, b)\). Here, since \( f(-4) = -28 \) and \( f(-2) = 10 \), and they have different signs, it confirms the existence of at least one zero between \(-4\) and \(-2\).

Step 4: Confirm continuity

Ensure the function \( f(x) = x^3 - 9x \) is continuous on the interval \([-4, -2]\). Since polynomials are continuous everywhere, \( f(x) \) is continuous on this interval.

Key Concepts

Polynomial Functions

Polynomial functions are essential mathematical expressions composed of variables and coefficients, combined using operations like addition, subtraction, multiplication, and non-negative integer exponents. An example you might encounter could be \( f(x) = x^3 - 9x \). This is a third-degree polynomial because the highest exponent of \( x \) is three.

Polynomials are described by their degree, which is determined by the highest power of the variable in the expression. They can take on various forms, each representing unique behaviors and properties. In general, polynomial functions are denoted as \( f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where each \( a_i \) is a coefficient and \( n \) is a non-negative integer.

• They are smooth and continuous, meaning they don't have any breaks or sharp corners.
• Graphs of polynomial functions can have turning points, and the number of these points can be up to one less than the degree of the polynomial.
• Polynomials can be used to model various real-world situations, making them very practical in sciences and engineering.

Understanding these characteristics is vital in comprehending the relationships and functions polynomials can depict.

Continuous Functions

A continuous function is a type of function where small changes in the input result in small changes in the output. More intuitively, a function is considered continuous if you can draw its entire graph without taking your pen off the paper. This features prominently in calculus, where it simplifies analysis and problem-solving significantly.

Polynomials, like the one given in our original exercise \( f(x) = x^3 - 9x \), are inherently continuous everywhere because they are made up of continuous operations—addition, subtraction, and multiplication of terms.

• Continuous functions don't have gaps or jumps in their graphs, which means the function's real-life interpretation is often consistent and predictable.
• To affirm continuity at any interval, polynomials require no further proof due to their non-breaking nature.

Continuity is a crucial property when applying the Intermediate Value Theorem, as only continuous functions on a closed interval can guarantee an intermediate value.

Zero of a Function

The zero of a function refers to the value of \( x \) where the function evaluates to zero, \( f(x) = 0 \). In other words, it's the point where the function crosses or touches the x-axis in a graph. Finding zeros is an important task, as it helps in understanding the roots or solutions of an equation.

In polynomial functions, zeros signify the inputs for which the polynomial yields zero as the output. For our specific polynomial function \( f(x) = x^3 - 9x \), applying the Intermediate Value Theorem helps us confirm the existence of a zero within a defined interval—between \( x = -4 \) and \( x = -2 \).

• Each zero corresponds to a solution of the equation set by equating the polynomial to zero, \( f(x) = 0 \).
• Determining zeros involves evaluating and sometimes solving the equation through factoring or applying numerical methods.

Understanding where and why these zeros occur aids not only in graphing functions but also in solving real-world problems that can be modeled by polynomial expressions.