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

Short answer

Domain: \((-\infty, \infty)\). Critical numbers: \(x = 0, -2\). Max at \(-2\), min at \(0\).

Step-by-step solution

Determine the Domain

The domain of a function encompasses all permissible input values, i.e., the values for which the function is defined. As the function given is a polynomial, specifically \(f(x) = x^3 + 3x^2 + 3\), it is defined for all real numbers. Thus, the domain of \(f\) is \((-\infty, \infty)\).

Find the Critical Numbers

To find the critical numbers, we need to find the first derivative of \(f\) and set it equal to zero. The first derivative of \(f(x) = x^3 + 3x^2 + 3\) is \(f'(x) = 3x^2 + 6x\). Setting \(f'(x) = 0\) gives us the equation \(3x^2 + 6x = 0\). Factor out a \(3x\): \(3x(x + 2) = 0\). Solving \(3x = 0\) and \(x + 2 = 0\) gives the critical numbers \(x = 0\) and \(x = -2\).

Determine Intervals of Increase/Decrease

To determine where the function is increasing or decreasing, create a number line including the critical numbers \(x = -2\) and \(x = 0\). Evaluate \(f'(x)\) in each interval: - For \(x < -2\), select \(x = -3\): \(f'(-3) = 3(-3)^2 + 6(-3) = 27 - 18 = 9\), positive, so \(f\) is increasing.- For \(-2 < x < 0\), select \(x = -1\): \(f'(-1) = 3(-1)^2 + 6(-1) = 3 - 6 = -3\), negative, so \(f\) is decreasing.- For \(x > 0\), select \(x = 1\): \(f'(1) = 3(1)^2 + 6(1) = 3 + 6 = 9\), positive, so \(f\) is increasing again.

Use the First Derivative Test

The First Derivative Test involves analyzing the sign change of \(f'(x)\) around each critical number to identify extremum points:- At \(x = -2\): The derivative changes from positive to negative as \(x\) crosses from \(-2\) to the right, indicating a local maximum at \(x = -2\).- At \(x = 0\): The derivative changes from negative to positive as \(x\) passes through \(0\), indicating a local minimum at \(x = 0\).

Key concepts

First Derivative Test

The First Derivative Test is a handy tool in calculus to determine the nature of critical points in a function. Let's break it down. When you take the derivative of a function, you essentially measure its slope or the rate of change. A critical point occurs when this slope is zero, meaning the function is neither increasing nor decreasing at that point. To find these critical points in our example function, we first took the derivative, which resulted in the expression \(3x^2 + 6x\). Setting this equal to zero and solving gives us the critical numbers \(x = 0\) and \(x = -2\).

Now, to apply the First Derivative Test, assess how the sign of the derivative changes around these points. If the derivative goes from positive to negative at a critical point, it suggests a local maximum. Conversely, if it changes from negative to positive, you're looking at a local minimum. In our example, the function showed a positive to negative shift at \(x = -2\), showing that point as a maximum, and a negative to positive shift at \(x = 0\), indicating a minimum.

Increasing and Decreasing Functions

Understanding when a function is increasing or decreasing is crucial for sketching its graph and analyzing its behavior. A function is said to be increasing on an interval if as \(x\) moves from left to right, the \(f(x)\) values get larger. Conversely, it's decreasing if the \(f(x)\) values get smaller as \(x\) increases.

For the function \(f(x) = x^3 + 3x^2 + 3\), we used its derivative \(f'(x) = 3x^2 + 6x\) to understand these trends. By testing the sign of the derivative in the intervals determined by the critical points (\(-\infty, -2\), \(-2, 0\), and \(0, \infty\)), we found:

• In the interval \(x < -2\), where we tested \(x = -3\), the derivative is positive, indicating a rising behavior.
• Between \(-2\) and \(0\), tested with \(x = -1\), the derivative is negative, showing a downward trend.
• For \(x > 0\), with \(x = 1\), the derivative becomes positive again, confirming the function ascends.

Thus, the function increases on \((-\infty, -2)\) and \((0, \infty)\), while decreasing on the interval \((-2, 0)\).

Polynomial Functions

Polynomial functions are one of the most fundamental types of functions in mathematics. They are expressions formed from sums of terms consisting of variables raised to whole-number exponents. Functions like \(f(x) = x^3 + 3x^2 + 3\) are cubic polynomials due to the highest degree of 3. These functions possess characteristics that make them easy to analyze, such as being defined and continuous everywhere on the real number line.

The simple operations involved in polynomials, such as addition, subtraction, and multiplication of terms, allow for straightforward differentiation to find critical points and analyze their behavior. Moreover, graphically, polynomials of degree \(n\) can have up to \(n-1\) critical points, which can be pivotal in deciding where the function increases or decreases. Their end behavior is dictated by the leading term, so in the long run, our example's term \(x^3\) means as \(x\) approaches positive or negative infinity, \(f(x)\) will rise or fall infinitely, respectively. This comprehensive understanding of polynomial functions lets us predict and describe the general shape and behavior of a graph with relative ease.