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Chapter 3: Problem 34

Use the Intermediate Value Theorem to show that each polynomial has a real zero between the given integers. \(f(x)=x^{3}-4 x^{2}+2 ;\) between 0 and 1

Short Answer

Expert verified
Using the Intermediate Value Theorem and considering that the polynomial \(f(x) = x^{3}-4 x^{2}+2\) is continuous everywhere and \(f(0)\) and \(f(1)\) have different signs, we can conclude that there is a real zero between 0 and 1.

Step by step solution

01

Evaluate at Endpoint

Evaluate the polynomial \(f(x)=x^{3}-4 x^{2}+2\) at \(x = 0\) and \(x = 1 \). This gives \(f(0) = 2\) and \(f(1)= 1 - 4 + 2=-1\)
02

Check Intermediate Value Theorem Conditions

To apply the Intermediate Value Theorem, we need to confirm that \(f(x)\) is continuous on the interval \([0, 1]\) and \(f(0)\) and \(f(1)\) have different signs.
03

Apply the Intermediate Value Theorem

Since \(f(x)\) is a polynomial, which is continuous everywhere, the first condition is met. The second condition is also met as \(f(0) = 2\) is positive whereas \(f(1) = -1\) is negative. Thus, by the Intermediate Value Theorem, there exists a number \(c\) in the interval (0, 1) such that \(f(c) = 0\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Polynomial Functions
Polynomial functions are mathematical expressions involving variables and coefficients. In simple terms, they are sums of multiple algebraic terms. Each term consists of a coefficient multiplied by a variable raised to a non-negative integer exponent. For example, the polynomial function \( f(x) = x^3 - 4x^2 + 2 \) consists of three terms: \( x^3 \), \(-4x^2 \), and \( 2 \).
Polynomials have a few notable features:
  • They are defined for all real numbers \( x \).
  • The degree of the polynomial is the highest power of the variable, which in \( f(x) = x^3 - 4x^2 + 2 \) is 3.
Polynomials are particularly important in mathematics because they are continuous and smooth functions which make them easier to analyze compared to other types of functions.
Real Zeros
A real zero of a polynomial function is a value of \( x \) where the function equals zero. In terms of the graph, this is where the curve intersects the x-axis. Determining these zeros is essential as they represent the roots of the equation.
Finding real zeros can be done through various methods:
  • Factoring the polynomial, if possible
  • Using graphing to visually identify intersections with the x-axis
  • Applying the Intermediate Value Theorem, which can identify the existence of zeros within an interval
In this exercise, the Intermediate Value Theorem helps us confirm that there is at least one real zero between 0 and 1 for the polynomial \( f(x) = x^3 - 4x^2 + 2 \).
Continuity
Continuity in a function means that the function's graph has no breaks, holes, or jumps. For polynomial functions, continuity is a given due to their algebraic structure.
Key aspects of continuity for polynomial functions include:
  • They are continuous over their entire domain, which is all real numbers.
  • There are no gaps or discontinuities in their graphs, making them smooth and predictable.
In our example, the function \( f(x) = x^3 - 4x^2 + 2 \) is continuous everywhere, including specifically on the interval \([0, 1]\). This continuity allows the application of the Intermediate Value Theorem to determine if a real zero exists within that interval.
Sign Change
Sign change in a polynomial function indicates a transition from positive to negative values, or vice versa. Detecting sign changes is crucial when using the Intermediate Value Theorem to find real zeros.
To effectively determine the sign change:
  • Evaluate the function at the endpoints of the interval of interest.
  • Observe the sign of the result: a change from positive to negative or negative to positive indicates a likely zero between those points.
For instance, in the case of the polynomial \( f(x) = x^3 - 4x^2 + 2 \), evaluating at \( x = 0 \) yields a positive 2, and at \( x = 1 \) yields a negative -1. This sign change confirms that there exists at least one real zero between the integers 0 and 1, aligning with the application of the Intermediate Value Theorem.

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