Problem 247 - 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. $$ f(x)=x^{3}-9 x, \quad \text { between } x=2 \text { and } x=4 $$

Step-by-Step Solution

Answer

There is a zero of the polynomial within the interval \((2, 4)\).

Step 1: Understand the Intermediate Value Theorem

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

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

Calculate \( f(2) \) and \( f(4) \) using the given polynomial \( f(x) = x^3 - 9x \). \[ f(2) = 2^3 - 9 imes 2 = 8 - 18 = -10 \] \[ f(4) = 4^3 - 9 imes 4 = 64 - 36 = 28 \]

Step 3: Check for Opposite Signs

Observe the signs of \( f(2) \) and \( f(4) \). - \( f(2) = -10 \), which is negative.- \( f(4) = 28 \), which is positive. Since they have opposite signs, the Intermediate Value Theorem applies.

Step 4: Conclude That There is a Zero in the Interval

Since \( f(2) \) and \( f(4) \) have opposite signs, according to the Intermediate Value Theorem, there must be at least one zero in the interval \((2, 4)\). This confirms the presence of a root.

Key Concepts

Polynomial Functions

Polynomial functions are fundamental in mathematics due to their versatile and wide-ranging applications. These functions are expressions involving multiple terms with varying powers of a single variable, typically denoted as \( x \). In general, a polynomial function can be expressed as:\[f(x) = a_n x^n + a_{n-1} x^{n-1} + \ldots + a_1 x + a_0\]where \( a_n, a_{n-1}, \ldots, a_0 \) are constants known as coefficients, and \( n \) is a non-negative integer representing the highest degree of the polynomial. The degree of the polynomial determines its complexity and the number of roots it can have. Here are some key points about polynomial functions:

• The highest power of \( x \) dictates the polynomial's degree.
• They can describe curves that cross the x-axis at multiple points, depending on the degree.
• Polynomials are smooth and continuous, meaning they don't have breaks or gaps.

Polynomial functions like \( f(x) = x^3 - 9x \) can have roots, which are values of \( x \) where \( f(x) = 0 \). Understanding how to find these roots is crucial.

Root Finding

Root finding is a central task when dealing with polynomial functions, as it involves identifying the values of \( x \) that make the polynomial equal to zero. These values are known as the roots or zeros of the polynomial. In the case of the polynomial \( f(x) = x^3 - 9x \), root finding focuses on discovering the x-values within a given interval where the function changes sign, indicating the presence of a root. Here's how you can approach this:

• Use methods like factoring or synthetic division for simpler polynomials.
• Apply numerical methods like the Intermediate Value Theorem for intervals with clear sign changes.

The Intermediate Value Theorem becomes particularly useful as it provides assurance of a zero between two points when the function crosses the x-axis. By evaluating the polynomial at the interval's endpoints and checking for sign changes, you can confirm the presence of at least one root.

Continuous Functions

Continuous functions are smooth curves on a graph without any breaks or jumps. This means you can draw them without lifting your pen from the paper. For polynomial functions, continuity is guaranteed across all real numbers, thanks to their nature as sums of powers of \( x \).Continuity plays a crucial role in applying the Intermediate Value Theorem. For the theorem to work, our function needs to be continuous over the closed interval \([a, b]\). Here's why:

• Continuous functions don't skip values, allowing you to infer changes without missing points.
• If a continuous function has different signs at the ends of an interval, it must cross the x-axis somewhere between those points.

In our example, the polynomial \( x^3 - 9x \) is continuous everywhere. Evaluating it at the endpoints \( x = 2 \) and \( x = 4 \) results in different signs, confirming through continuity and the Intermediate Value Theorem, the presence of a root in that interval.