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. \(f(x)=2 x^{3}+x^{2}-x+3\)

Short answer

Domain of \(f\): \(\mathbb{R}\). Critical numbers: \(x = \frac{-1 \pm \sqrt{7}}{6}\). Using the derivative test, check intervals for increase/decrease and relative extrema.

Step-by-step solution

Determining the Domain

The domain of a polynomial function is all real numbers, since there are no restrictions like division by zero or square roots of negative numbers. Thus, the domain of \( f(x) = 2x^3 + x^2 - x + 3 \) is \( \mathbb{R} \), which means it is any real number.

Finding the Critical Numbers

To find the critical numbers of \( f(x) \), we first need to calculate its derivative \( f'(x) \). The derivative of \( f(x) = 2x^3 + x^2 - x + 3 \) is \( f'(x) = 6x^2 + 2x - 1 \). Set the derivative equal to zero and solve for \( x \): \( 6x^2 + 2x - 1 = 0 \). This can be solved using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 6, b = 2, c = -1 \). Solving, we find the critical numbers are \( x = \frac{-1 \pm \sqrt{7}}{6} \).

Creating a Number Line for Increasing/Decreasing Intervals

Place the critical numbers \( x = \frac{-1 + \sqrt{7}}{6} \) and \( x = \frac{-1 - \sqrt{7}}{6} \) on a number line. Choose test points in each interval defined by these critical numbers: use \( x = -1 \) for \( x < \frac{-1 - \sqrt{7}}{6} \), \( x = 0 \) for \( \frac{-1 - \sqrt{7}}{6} < x < \frac{-1 + \sqrt{7}}{6} \), and \( x = 1 \) for \( x > \frac{-1 + \sqrt{7}}{6} \). Evaluate \( f'(x) \) at these points to determine the sign. If \( f'(x) > 0 \), the function is increasing, and if \( f'(x) < 0 \), the function is decreasing.

Applying the First Derivative Test

Use the signs from Step 3 to determine the nature of each critical point. If \( f'(x) \) changes from negative to positive at a critical point, it indicates a relative minimum. If \( f'(x) \) changes from positive to negative, it indicates a relative maximum. Check the sign changes around \( x = \frac{-1 + \sqrt{7}}{6} \) and \( x = \frac{-1 - \sqrt{7}}{6} \) to see if they are relative maxima or minima.

Key concepts

Critical Numbers

Critical numbers of a function help us understand where the function's graph may have high or low points, also known as peaks or troughs. To identify these critical numbers, we first need to calculate the derivative of the function. The derivative, noted as \( f'(x) \), represents the rate of change of the function. In our problem, we start with the polynomial function \( f(x) = 2x^3 + x^2 - x + 3 \).
Computing the derivative gives us \( f'(x) = 6x^2 + 2x - 1 \). Critical numbers occur where the derivative is zero or undefined. Since the derivative is a polynomial, it is always defined, so we only need to find where \( f'(x) = 0 \).
Solving the equation \( 6x^2 + 2x - 1 = 0 \) using the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), reveals the critical numbers: \( x = \frac{-1 \pm \sqrt{7}}{6} \). These are the x-values where the graph might have peaks or valleys.

Increasing and Decreasing Intervals

Once we have the critical numbers, we can use them to find where the function is increasing or decreasing. This involves testing the sign of the derivative \( f'(x) \) in the intervals created by these critical numbers. In this exercise, we identified the critical numbers as \( x = \frac{-1 + \sqrt{7}}{6} \) and \( x = \frac{-1 - \sqrt{7}}{6} \).
We set up our number line with these points and choose test points in the intervals they define. For instance, pick \( x = -1 \) for the interval left of \( \frac{-1 - \sqrt{7}}{6} \), \( x = 0 \) for the interval between \( \frac{-1 - \sqrt{7}}{6} \) and \( \frac{-1 + \sqrt{7}}{6} \), and \( x = 1 \) for the interval right of \( \frac{-1 + \sqrt{7}}{6} \).
Evaluate the derivative at these test points:

• If \( f'(x) > 0 \), the function is increasing in that interval.
• If \( f'(x) < 0 \), the function is decreasing.

First Derivative Test

The First Derivative Test helps to determine the nature of critical points found in the previous steps. Specifically, it tells us whether these points are relative maxima, minima, or neither. We use the results from the sign test of \( f'(x) \) over each interval.
Here's how it works:

• If \( f'(x) \) changes from negative to positive as we pass a critical number from left to right, the function has a relative minimum at that point.
• If \( f'(x) \) changes from positive to negative, the function has a relative maximum.

Using the critical numbers \( x = \frac{-1 + \sqrt{7}}{6} \) and \( x = \frac{-1 - \sqrt{7}}{6} \), examine these transitions:
Determine the sign changes of \( f'(x) \) across these values to confirm whether they are relative maxima or minima.

Polynomial Function Domain

Determining the domain of a polynomial function is straightforward, as these functions have no restrictions, such as division by zero or taking the square root of a negative number. This means polynomial functions are typically defined for all real numbers.
In this case, we examine \( f(x) = 2x^3 + x^2 - x + 3 \). Because it’s a polynomial, its domain is \( \mathbb{R} \). Simply put, you can plug any real number into this function, and it will yield a real output. This universal domain makes polynomial functions very versatile in calculus.