Problem 19 - Thomas Calculus in SI Units

Question

Show that the functions that have exactly one zero in the given interval. $$f(x)=x^{4}+3 x+1, \quad[-2,-1]$$

Step-by-Step Solution

Answer

The function has exactly one zero in the interval \([-2, -1]\).

Step 1: Verify the continuity of the function

The function \( f(x) = x^4 + 3x + 1 \) is a polynomial function, which is continuous everywhere. Hence, it is continuous in the interval \([-2, -1]\).

Step 2: Evaluate the function at the interval boundaries

Calculate \( f(-2) \) and \( f(-1) \).\[ f(-2) = (-2)^4 + 3(-2) + 1 = 16 - 6 + 1 = 11 \]\[ f(-1) = (-1)^4 + 3(-1) + 1 = 1 - 3 + 1 = -1 \]Thus, \( f(-2) = 11 \) and \( f(-1) = -1 \).

Step 3: Apply the Intermediate Value Theorem (IVT)

Since \( f(x) \) is continuous and \( f(-2) > 0 \) while \( f(-1) < 0 \), the Intermediate Value Theorem tells us there must be at least one zero in \((-2, -1)\) where the function changes sign.

Step 4: Check the uniqueness of the zero

To confirm uniqueness, let's examine the function's behaviour.Calculate the derivative: \[ f'(x) = 4x^3 + 3 \]The derivative is always positive for \( x \in [-2, -1] \) because \( 4x^3 + 3 > 0 \) due to the dominant \( +3 \). Therefore, \( f(x) \) is strictly increasing on \([-2, -1]\).Since the function is strictly increasing and crosses the x-axis only once, it has exactly one zero in the interval.

Key Concepts

Polynomial Function

A polynomial function is a type of mathematical expression that involves sums of powers of variables, each multiplied by coefficients. The general form of a polynomial function is:

• a_n \cdot x^n + a_{n-1} \cdot x^{n-1} + ... + a_1 \cdot x + a_0

where:

• \(x\) is the variable,
• \(a_n, a_{n-1}, ..., a_0\) are the coefficients, and
• \(n\) is the highest power, known as the degree of the polynomial.

Polynomial functions are considered continuous everywhere on the real number line. This means that you can draw the graph of a polynomial without lifting your pen from the paper, indicating no breaks or holes.
In our exercise, the given function \(f(x) = x^4 + 3x + 1\) is a polynomial function with a degree of 4, since the highest power of \(x\) is 4. It is continuous by nature, which is crucial for applying the Intermediate Value Theorem later.

Continuity

Continuity is a fundamental concept in mathematics, especially when dealing with functions like polynomials. In simple terms, a function is said to be continuous on an interval if you can draw it without taking your pen off the paper.
Mathematically, for a function to be continuous at a point \(c\), it must satisfy three conditions:

• The function \(f(x)\) must be defined at \(x = c\).
• The limit of \(f(x)\) as \(x\) approaches \(c\) from both sides must exist.
• The limit must equal \(f(c)\).

For polynomial functions, these criteria are always met, ensuring they are continuous everywhere.
In the given exercise, the function \(f(x) = x^4 + 3x + 1\) is continuous on the interval \([-2, -1]\) without any interruptions. This property is crucial when applying the Intermediate Value Theorem to show that the function has at least one zero in the interval.

Strictly Increasing Function

A strictly increasing function is a type of function where the output value increases as the input value increases across a certain interval.
Formally, if \(f(x)\) is strictly increasing on an interval, then for any two numbers \(x_1\) and \(x_2\) in that interval, if \(x_1 < x_2\), it must be that \(f(x_1) < f(x_2)\). This means the function never dips or flattens out; it steadily climbs as you move from left to right on the graph.

• The derivative of the function, when positive, is a strong indicator of a strictly increasing function.

In the task at hand, we found the derivative of the function \(f(x) = x^4 + 3x + 1\), which is \(f'(x) = 4x^3 + 3\). Notably, this derivative is positive over the interval \([-2, -1]\), confirming that \(f(x)\) is strictly increasing in this region.
This characteristic helps assert that the function crosses the x-axis only once, verifying that it has exactly one zero, which is especially useful for determining the uniqueness of the zero in the specified interval.