Problem 27 - 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. \(f(x)=-x^{4}+4,\) between \(x=1\) and \(x=3\).

Step-by-Step Solution

Answer

There is at least one zero of \( f(x) = -x^4 + 4 \) in the interval \([1, 3]\).

Step 1: Understand the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem (IVT) states that for any continuous function \( f \), if \( f(a) \) and \( f(b) \) have opposite signs, there exists at least one \( c \) in the interval \([a, b]\) such that \( f(c) = 0 \). Here, we need to check if \( f(x) = -x^4 + 4 \) satisfies these conditions over the interval \([1, 3]\).

Step 2: Calculate \( f(1) \)

Substitute \( x = 1 \) into the function:\[ f(1) = -(1)^4 + 4 = -1 + 4 = 3 \]So, \( f(1) = 3 \).

Step 3: Calculate \( f(3) \)

Substitute \( x = 3 \) into the function:\[ f(3) = -(3)^4 + 4 = -81 + 4 = -77 \]So, \( f(3) = -77 \).

Step 4: Check for Opposite Signs

Note that \( f(1) = 3 \) is positive, and \( f(3) = -77 \) is negative. Since \( f(1) \) and \( f(3) \) have opposite signs, by the IVT, there must be at least one zero within the interval \([1, 3]\).

Step 5: Conclusion

By the Intermediate Value Theorem, the continuous function \( f(x) = -x^4 + 4 \) changes sign between \( x = 1 \) and \( x = 3 \). Thus, there is at least one zero in the interval \([1, 3]\).

Key Concepts

Polynomials

Polynomials are algebraic expressions that consist of variables raised to non-negative integer powers, combined by addition, subtraction, and multiplication. They can take various forms, but a simple form like our example is a single-variable polynomial:

• The general form of a polynomial is: \( a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0 \), where each \(a_i\) is a coefficient.
• Polynomials are classified based on their degree, which is the highest power of the variable. In our example, \(-x^4 + 4\), it's a degree 4 polynomial.
• Understanding polynomials is key in various areas of math, such as calculus and algebra. They are used to describe curves and surfaces.

They can describe real-world phenomena and are foundational in studies of motion, growth, and decay.
Studying zeroes of polynomials enables us to find solutions to equations set to zero, which is fundamental in many mathematical applications, both theoretical and practical.

Continuous Functions

A continuous function is one where small changes in the input (x-value) produce small changes in the output (y-value). This smoothness is essential for many mathematical theorems, including the Intermediate Value Theorem (IVT). Here's what you need to know about continuous functions:

• Continuity means that the graph of the function is an unbroken line; there are no gaps or holes.
• Mathematically, a function \( f(x) \) is continuous at a point \( x = c \) if \( \lim_{x \to c} f(x) = f(c) \).
• In the given function \(-x^4 + 4\), the continuity is due to the fact that all polynomial functions, such as this one, are continuous everywhere.

In short, continuous functions allow us to apply powerful mathematical concepts to find and analyze roots within intervals, which is what the IVT focuses on.

Zeros of a Function

Zeros of a function are the values of \( x \) for which \( f(x) = 0 \). These points are significant because they represent where the graph of a function intersects the x-axis. Determining these zeros helps solve equations and understand the behavior of functions. Here's how we look at zeros:

• In our example, identifying zeros is crucial as they indicate where the polynomial crosses the x-axis in the interval \([1, 3]\).
• The Intermediate Value Theorem helps us locate at least one zero in the interval, since there is a sign change from \( f(1) = 3 \) to \( f(3) = -77 \).
• Finding zeros is a fundamental aspect of calculus and algebra, used in solving equations, optimizing functions, and conducting analyses.

Recognizing and calculating the zeros of a polynomial provides insight into its structure and solutions, forming the basis for much of mathematical problem-solving.