Problem 24 - College Algebra and Calculus: An Applied Approach

Question

Use the Intermediate Value Theorem to show that the function has at least one zero in the interval \([a, b] .\) (You do not have to approximate the zero.) $$f(x)=x^{5}-3 x+3, \quad[-2,-1]$$

Step-by-Step Solution

Answer

Applying the Intermediate Value Theorem, it is shown that the function \(f(x) = x^5 -3x + 3\) has at least one root in the interval \([-2, -1]\).

Step 1: Evaluate the function at the interval endpoints

First, we would start by calculating the function \(f\) at the endpoints of the interval \([-2, -1]\). This would give us \(f(-2)\) and \(f(-1)\). \(f(-2) = (-2)^5 - 3*(-2) + 3 = -32 + 6 + 3 = -23\)\(f(-1) = (-1)^5 - 3*(-1) + 3 = -1 + 3 + 3 = 5\)

Step 2: Apply the Intermediate Value Theorem

According to the Intermediate Value Theorem, since the function is continuous in the interval \([-2, -1]\) (which is true for the given function as all polynomial functions are continuous everywhere), and \(0\) lies between \(f(-2) = -23\) and \(f(-1) = 5\), then there exists a number \(c\) in the interval \([-2, -1]\) such that \(f(c) = 0\).

Step 3: Conclusion

Hence, it is shown that the function \(f(x) = x^5 -3x + 3\) has at least one zero in the interval \([-2, -1]\).

Key Concepts

Understanding the Zero of a Function

The concept of a zero of a function is critical when exploring mathematical functions. A zero or root of a function is the value of the input \(x\) where the output or the function value is zero. In simpler terms, it's the point where the graph of the function intersects the x-axis. Finding zeros of a function is essential because it helps in understanding the behavior of the function, predicting outcomes, and solving equations.

• If \(f(c) = 0\) for some value \(c\), then \(c\) is a zero of the function \(f\).
• Zeros are crucial in various applications, such as optimization problems and graphing functions.

When dealing with polynomial functions, finding zeros can often give insights into the roots of complex equations.

Exploring Continuous Functions

A continuous function is a function that has no breaks, jumps, or interruptions in its graph. This means that you can draw the graph of a continuous function without lifting your pencil from the paper. In the context of the Intermediate Value Theorem, continuity plays a vital role because the theorem applies only to continuous functions.

• For a function to be continuous at a point \(c\), the limit as \(x\) approaches \(c\) must equal the function value \(f(c)\).
• Polynomial functions, like the one provided in the exercise, are continuous everywhere because they are constructed from basic operations (addition, multiplication, and exponentiation) that are continuous.

Understanding continuity helps us predict and confirm behavior such as the existence of zeros, as is the case with the given polynomial function on the specified interval.

Why Polynomial Functions Are Key

Polynomial functions are a class of functions that involve powers of \(x\) with real coefficients. They are important in mathematics due to their versatility and simple continuity. Polynomial functions, which include the expression \(x^5 - 3x + 3\) in the exercise, are defined for all real numbers and have graphs that are smooth and unbroken.

• Characteristics of polynomials include having terms constructed by \ multiplication, addition, and real-number coefficients, leading to them being continuous and easy to integrate or differentiate.
• In the case of the given function, it is a degree 5 polynomial, which typically implies five possible real or complex roots (although not necessarily all distinct or real).

The continuity and differentiability of polynomial functions make them a popular choice for demonstrating the Intermediate Value Theorem and similar concepts. They assure the presence of zeros within specified intervals when certain conditions are met.