Problem 10 - Precalculus: Functions and Graphs

Question

Use the Intermediate value theorem to show that \(f\) has a zero between \(a\) and \(b\) $$\begin{array}{c} f(x)=x^{5}-3 x^{4}-2 x^{3}+3 x^{2}-9 x-6 \\ a=3, \quad b=4 \end{array}$$

Step-by-Step Solution

Answer

The function has a zero between 3 and 4.

Step 1: Understanding the Function

The function given is a polynomial: \( f(x) = x^5 - 3x^4 - 2x^3 + 3x^2 - 9x - 6 \). To apply the Intermediate Value Theorem, we first need to evaluate this function at the endpoints \( a = 3 \) and \( b = 4 \).

Step 2: Evaluating the Function at the Endpoints

Calculate \( f(3) \) and \( f(4) \).\[ f(3) = 3^5 - 3 \cdot 3^4 - 2 \cdot 3^3 + 3 \cdot 3^2 - 9 \cdot 3 - 6 \]\[ = 243 - 243 - 54 + 27 - 27 - 6 \]\[ = -60 \]Next, calculate \( f(4) \):\[ f(4) = 4^5 - 3 \cdot 4^4 - 2 \cdot 4^3 + 3 \cdot 4^2 - 9 \cdot 4 - 6 \]\[ = 1024 - 768 - 128 + 48 - 36 - 6 \]\[ = 134 \]

Step 3: Check the Sign Change

Evaluate the results from the previous step: \( f(3) = -60 \) and \( f(4) = 134 \). There is a sign change from negative to positive, indicating that there must be at least one root between \( x = 3 \) and \( x = 4 \).

Step 4: Applying the Intermediate Value Theorem

According to the Intermediate Value Theorem, if \( f \) is continuous on \([a, b]\) and \( N \) is any number between \( f(a) \) and \( f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f(c) = N \). In this case, since \( N = 0 \) is between \( f(3) = -60 \) and \( f(4) = 134 \), there must be a value \( c \) between 3 and 4 where \( f(c) = 0 \).

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions consisting of variables and coefficients connected by addition, subtraction, and multiplication. Each term in a polynomial has a variable raised to a non-negative integer exponent, like in the function \( f(x) = x^5 - 3x^4 - 2x^3 + 3x^2 - 9x - 6 \). The highest exponent, in this case 5, defines the degree of the polynomial. Some characteristics of polynomial functions include:

• Continuity: Polynomials are continuous over the real numbers, which means they have no breaks, holes, or gaps in their graphs. This is crucial when applying the Intermediate Value Theorem as continuity ensures that the function behaves predictably within intervals.
• Smooth Curves: Polynomials create smooth, flowing curves without sharp turns, further emphasizing their predictable behavior on a graph.
• Roots or Zeros: These are the points where the polynomial equals zero. Finding these is often essential for solving equations.

Understanding these properties allows us to better analyze polynomial behaviors, especially when assessing roots within specific intervals.

Root Finding

Root finding is a fundamental concept in algebra that aims to determine where a function crosses the x-axis, known as the function's zeros. For the polynomial \( f(x) \), a root or zero is a solution to \( f(x) = 0 \). Methods used for root finding include:

• Graphical Method: Plotting the function and observing where it intersects the x-axis. This visual representation helps identify approximate root locations.
• Algebraic Techniques: Applying factorization or synthetic division to solve equations directly when possible.
• Numerical Methods: Techniques such as the Intermediate Value Theorem, which leverages the continuity of polynomial functions in specific intervals to indicate where roots may lie.

By assessing the change in sign of \( f(x) \) at the boundaries, like \( f(3) = -60 \) and \( f(4) = 134 \), we infer that the polynomial must cross zero between \( x = 3 \) and \( x = 4 \). Such insights are crucial for deeper mathematical problem-solving.

Continuous Functions

Continuous functions are those that are uninterrupted and seamless at every point within an interval. For polynomials like \( f(x) = x^5 - 3x^4 - 2x^3 + 3x^2 - 9x - 6 \), continuity is inherent due to their foundational structure. Features of continuous functions include:

• No Gaps: A continuous function will have no breaks in its graph within its domain, ensuring consistent behavior across any interval.
• Ability to Apply Theorems: The Intermediate Value Theorem specifically relies on a function's continuity. Because polynomials are inherently continuous, we can confidently apply this theorem to determine if a zero exists within a given interval when the function's values change signs.
• Smooth Curves: Continuous functions result in smooth, non-disjoint graphs.

This consistent nature of polynomials allows us to use the Intermediate Value Theorem to confirm root existence, effectively applying it in scenarios such as those provided in the exercise.