Question

Use the Intermediate Value Theorem to prove that \(x^{3}+3 x-2=0\) has a real solution between 0 and 1 .

Short answer

The equation has a real root between 0 and 1.

Step-by-step solution

Define f(x)

First, define the function given in the equation. We have the function:\[ f(x) = x^3 + 3x - 2 \]

Calculate f(0)

Substitute \( x = 0 \) into the function to find \( f(0) \).\[ f(0) = (0)^3 + 3(0) - 2 = -2 \]

Calculate f(1)

Substitute \( x = 1 \) into the function to find \( f(1) \).\[ f(1) = (1)^3 + 3(1) - 2 = 1 + 3 - 2 = 2 \]

Verify Function Continuity

The function \( f(x) = x^3 + 3x - 2 \) is a polynomial. Polynomials are continuous over all real numbers, so \( f(x) \) is continuous on the interval \([0, 1]\).

Apply the Intermediate Value Theorem

The Intermediate Value Theorem states that if \( f(x) \) is continuous on \([a, b]\), and \( f(a) \) and \( f(b) \) have opposite signs, then there exists a \( c \) in \((a, b)\) such that \( f(c) = 0 \).Since \( f(0) = -2 \) and \( f(1) = 2 \), and they have opposite signs, by the Intermediate Value Theorem, there must be a \( c \) in \((0, 1)\) such that \( f(c) = 0 \).

Key concepts

Understanding Polynomials

A polynomial is a type of mathematical expression that involves a sum of powers in one or more variables multiplied by coefficients. In simpler terms, it's like a combination of numbers and letters that represent specific values. When you think of a polynomial, imagine it as a neat sum of terms built around powers of the variable, usually presented as:

• Terms: Each term in a polynomial is a part of the expression, such as \(x^3\), \(3x\), or a constant like \(-2\).
• Variables and Coefficients: The letters (usually \(x\)) represent the variables, while numbers in front of these letters (such as 1 in \(x^3\) or 3 in \(3x\)) are the coefficients.
• Degree: The degree of the polynomial is determined by the highest power of the variable, which in this case is 3, making the expression \(x^3 + 3x - 2\) a cubic polynomial.

Polynomials have special properties, especially related to their graphs. They are smooth and continuous, which leads us to our next discussion about continuity.

The Principle of Continuity in Polynomials

Continuity is a vital concept in mathematics, especially when dealing with polynomials. A function is continuous if you can draw it on a graph without lifting your pencil. For polynomials, this quality is inherently present.

• How Continuity Works: For a polynomial like \(f(x) = x^3 + 3x - 2\), continuity means that you can trace the graph smoothly from left to right without any jumps, breaks, or gaps. This unbroken path assures that the function doesn't suddenly disappear and reappear elsewhere along the graph.
• Why It Matters: The beautifully predictable nature of polynomials allows us to apply important mathematical theorems like the Intermediate Value Theorem. These theorems rely on the reliability of continuous functions to work effectively.
• Interval Consideration: In our given exercise, analyzing the interval \([0, 1]\) is crucial because it ensures that the continuity property holds between these two points. This lets us explore further for solutions within this range.

Understanding the seamless flow of polynomials through continuity paves the way for applying more complex mathematical ideas, such as finding solutions or roots.

Finding Real Solutions with the Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a powerful mathematical tool used to prove the existence of real solutions in a certain interval for continuous functions like polynomials.

• The Core Idea: The theorem states that if a function is continuous on a closed interval \([a, b]\), and the values of the function at the endpoints, \(f(a)\) and \(f(b)\), have opposite signs, then there must be at least one real solution between \(a\) and \(b\). In simple terms, the graph of the function has to cross the x-axis at some point in that interval.
• Applying IVT: For the function \(f(x) = x^3 + 3x - 2\), checking \(f(0) = -2\) and \(f(1) = 2\) tells us that the function changes from negative to positive as \(x\) goes from 0 to 1. This change in sign is a guarantee that the curve crosses the x-axis between these points, ensuring there is a root or solution there.
• Reassurance of Real Solutions: Thanks to the IVT, students can be confident that a real solution exists within the interval \((0, 1)\), even if the exact value isn't calculated. This insight is handy in both theoretical and practical applications, laying a foundation for exploring more complex problem-solving techniques.

By utilizing IVT, we bridge the abstract world of equations with the concrete reality of solution points on a graph, illuminating the path to deeper understanding and discovery in mathematics.