Question

Use the Intermediate Value Theorem to show that each polynomial function has a real zero in the given interval. $$ f(x)=3 x^{3}-10 x+9 ;[-3,-2] $$

Short answer

Yes, there is a real zero in the interval \([-3, -2]\).

Step-by-step solution

- Understand the Intermediate Value Theorem

The Intermediate Value Theorem states that if a continuous function, say \(f(x)\), takes on different signs at two points \(a\) and \(b\) in an interval \([a, b]\), then there exists at least one point \(c\) in \((a, b)\) where \(f(c) = 0\).

- Check Continuity of the Polynomial Function

Check if the given polynomial function \(f(x) = 3x^3 - 10x + 9\) is continuous on the interval \([-3, -2]\). Polynomials are continuous everywhere, so \(f(x)\) is continuous on \([-3, -2]\).

- Evaluate the Function at the Endpoints of the Interval

Calculate \(f(-3)\) and \(f(-2)\). These points will help determine if there is a sign change in the function values.\[f(-3) = 3(-3)^3 - 10(-3) + 9 = 3(-27) + 30 + 9 = -81 + 30 + 9 = -42\]\[f(-2) = 3(-2)^3 - 10(-2) + 9 = 3(-8) + 20 + 9 = -24 + 20 + 9 = 5\]

- Analyze the Signs of the Function Values at Endpoint

After computing, we find:\[f(-3) = -42\] \[f(-2) = 5\]Since \(f(-3)\) is negative and \(f(-2)\) is positive, there is a sign change.

- Apply the Intermediate Value Theorem

Since \(f(x)\) is continuous on \([-3, -2]\) and there is a sign change between \(f(-3)\) and \(f(-2)\), by the Intermediate Value Theorem, there exists some \(c\in[-3, -2]\) such that \(f(c) = 0\). Thus, the function \(f(x) = 3x^3 - 10x + 9\) has a real zero in the interval \([-3, -2]\).

Key concepts

Real Zero

The concept of a real zero refers to a point where a function crosses the x-axis, meaning the value of the function at this point is zero. We denote it as f(c) = 0. Finding real zeros of functions, especially polynomial functions, can help us understand the behavior and properties of the function. In the case of the Intermediate Value Theorem, identifying points where the function changes sign is essential in proving the existence of a real zero within an interval. For example, in the given polynomial function f(x) = 3x^3 - 10x + 9, evaluating the function at the interval endpoints (-3 and -2) revealed a sign change, indicating a real zero within this range according to the theorem.

Polynomial Function

A polynomial function is a function involving terms that are non-negative integer powers of a variable. It has the general form:

\[P(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0\]

where a_n, a_{n-1}, ..., a_1, a_0 are constants, and n is a non-negative integer. Polynomial functions can have various degrees, which tell us the highest power of the variable in the function. For instance, f(x) = 3x^3 - 10x + 9 is a polynomial function of degree 3 because the highest power of x is 3. These functions are continuous and smooth, which means they don't have breaks, jumps, or sharp corners. This continuity makes them ideal candidates for applying the Intermediate Value Theorem, as we did in the provided exercise.

Continuous Function

A continuous function is a function where small changes in the input result in small changes in the output, without any sudden jumps or breaks. Mathematically, a function f(x) is continuous at a point c if:

The function f(c) is defined.

The limit of f(x) as x approaches c exists.

The limit of f(x) as x approaches c equals f(c).

In the context of polynomial functions, they are inherently continuous because these functions are defined for all real numbers and don't contain any breaks or gaps. This characteristic is crucial when applying the Intermediate Value Theorem. Since the theorem relies on the continuity of functions, polynomials automatically satisfy this requirement, allowing us to analyze sign changes between intervals to find real zeros. By confirming that f(x) = 3x^3 - 10x + 9 is continuous on the interval [-3, -2], we could confidently apply the theorem to show there is a real zero in this interval.