Question

A function \(f(x)\) is given. (a) Give the domain of \(f\). (b) Find the critical numbers of \(f\). (c) Create a number line to determine the intervals on which \(f\) is increasing and decreasing. (d) Use the First Derivative Test to determine whether each critical point is a relative maximum, minimum, or neither. \(f(x)=x^{3}-3 x^{2}+3 x-1\)

Short answer

The domain is \((-\infty, \infty)\), critical number is \(x=1\), function increases on \((-\infty, \infty)\). No relative extrema at \(x=1\).

Step-by-step solution

Determine the Domain

The domain of a polynomial function is all real numbers. Since \(f(x) = x^3 - 3x^2 + 3x - 1\) is a polynomial, its domain is all real numbers: \((-\infty, \infty)\).

Find the Critical Numbers

To find critical numbers, we need to compute the derivative of \(f(x)\) and set it equal to zero. The derivative is given by: \[f'(x) = 3x^2 - 6x + 3.\]Set the derivative equal to zero:\[3x^2 - 6x + 3 = 0.\]This simplifies to:\[x^2 - 2x + 1 = 0\]which can be factored as:\[(x-1)^2 = 0.\]Solving \((x-1)^2 = 0\), we find the critical number to be \(x = 1\).

Create a Number Line for Intervals

We use the critical number(s) to divide the number line into intervals. Here, we have intervals: \((-\infty, 1)\) and \((1, \infty)\). We will test these intervals to determine where \(f(x)\) is increasing or decreasing.

Test Intervals for Increasing/Decreasing

Choose a test point from each interval and evaluate \(f'(x)\) to determine the sign:- For \((-\infty, 1)\), use \(x = 0\): \(f'(0) = 3(0)^2 - 6(0) + 3 = 3 > 0\), so \(f(x)\) is increasing.- For \((1, \infty)\), use \(x = 2\): \(f'(2) = 3(2)^2 - 6(2) + 3 = 3 > 0\), so \(f(x)\) is increasing.

Apply the First Derivative Test

Since \(f(x)\) is increasing on both sides of \(x = 1\), the critical point \(x = 1\) is not a maximum or minimum. Thus, \(x = 1\) is neither a relative maximum nor a relative minimum.

Key concepts

Polynomial Function

A polynomial function is an expression consisting of variables, coefficients, and the operations of addition, subtraction, multiplication, and non-negative integer exponents. Polynomials are very common in mathematics because they represent basic algebraic problems. The general form is given by:

• A general polynomial: \(a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0\)
• Where \(a_n\), \(a_{n-1}\), ... , \(a_0\) are constants and \(n\) is a non-negative integer.

For example, the function in the original problem, \(f(x) = x^3 - 3x^2 + 3x - 1\), is a polynomial function of degree 3, known as a cubic polynomial.
This function is continuous and smooth, meaning it has no breaks or sharp turns. This allows us to easily analyze its behavior across its domain.

Derivative Test

The derivative test, specifically the First Derivative Test, is used to determine where a function is increasing or decreasing, and to identify relative maxima or minima. The key steps are:

• Find the derivative \(f'(x)\).
• Determine critical numbers by setting \(f'(x) = 0\) or where \(f'(x)\) is undefined.
• Evaluate \(f'(x)\) around the critical numbers to check sign changes.

In the problem, the derivative is calculated as \(f'(x) = 3x^2 - 6x + 3\). Solving for zeros finds critical numbers. Using the test intervals formed by these numbers, you can determine where the function is increasing or decreasing.
This problem's critical number \(x = 1\) shows an increasing function on both sides, indicating neither a maximum nor a minimum.

Domain of a Function

The domain of a function represents all possible input values (\(x\)) for which the function is defined. For polynomial functions, like \(f(x) = x^3 - 3x^2 + 3x - 1\), the domain is typically all real numbers \((-\infty, \infty)\).
Polynomials don't have any restrictions, such as division by zero or negative square roots, which means they are defined for every real number. This makes them particularly straightforward to work with in calculus and algebra. Knowing the domain is essential before finding derivatives or evaluating the function's behavior.

Critical Numbers

Critical numbers are specific values of \(x\) in the domain where the derivative \(f'(x)\) is zero or undefined. These points are crucial in determining the behavior of a function because they indicate where potential maximums, minimums, or inflections occur.
In the problem, finding the derivative \(f'(x) = 3x^2 - 6x + 3\) and setting it to zero gives the equation \((x-1)^2 = 0\). Solving this, you find the critical number is \(x = 1\).
Understanding where these numbers are helps in summarizing the function's behavior, such as identifying where it increases or decreases. This concept is essential for graphing functions and analyzing complex functions efficiently.