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

Short answer

The domain of \(f\) is \((-\infty, \infty)\). The critical number is \(x = -1\), which is a relative minimum. \(f(x)\) is decreasing on \((-\infty, -1)\) and increasing on \((-1, \infty)\).

Step-by-step solution

Determine the Domain of f(x)

The function given is a polynomial, specifically a quadratic function, which is defined for all real numbers. Thus, the domain of the function \(f(x) = x^2 + 2x - 3\) is all real numbers. Therefore, the domain is given by: \((-\infty, \infty)\).

Find the Critical Numbers

To find the critical numbers of \(f(x)\), we first take its derivative. The function is \(f(x) = x^2 + 2x - 3\). Taking the derivative:\[f'(x) = 2x + 2\]Set \(f'(x) = 0\) to find critical numbers:\[2x + 2 = 0 \x = -1\]So, the critical number is \(x = -1\).

Number Line for Increasing and Decreasing Intervals

Create a number line and test intervals determined by the critical number \(x = -1\). Choose test points in the intervals \((-\infty, -1)\) and \((-1, \infty)\):1. Test point \(x = -2\) in \((-\infty, -1)\): \(f'(-2) = 2(-2) + 2 = -2\) \Rightarrow\ f'(x) < 0\ (decreasing)2. Test point \(x = 0\) in \((-1, \infty)\): \(f'(0) = 2(0) + 2 = 2\) \Rightarrow\ f'(x) > 0\ (increasing)Hence, \(f(x)\) is decreasing on \((-\infty, -1)\) and increasing on \((-1, \infty)\).

Apply the First Derivative Test

According to the first derivative test, the critical point \(x = -1\) is where \(f(x)\) changes from decreasing to increasing. Since the sign of the derivative changes from negative to positive at \(x = -1\), this indicates that \(x = -1\) is a relative minimum point for \(f(x)\).

Key concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (usually represented by "x") for which the function is defined. In simpler terms, it's what you can plug into the function without breaking any mathematical rules, like dividing by zero or taking the square root of a negative number.

For polynomial functions, like the quadratic function given in the exercise, the domain is particularly straightforward. Polynomials are defined for all real numbers because they don't have any restrictions or undefined points.

• For example, the domain of the polynomial function \(f(x) = x^2 + 2x - 3\) is all real numbers, which means you can use any number for x.
• This can be expressed as \((-\infty, \infty)\) in interval notation, indicating there are no gaps or limits to the values x can take.

Understanding the domain is fundamental because it sets the stage for further analysis on critical numbers and behavior of the function.

Critical Numbers

Critical numbers are particular values of x where the derivative of the function is zero or undefined. These numbers are essential because they pinpoint where the function's graph changes direction, which could lead to local maximums or minimums or nothing at all.

To find critical numbers:

• First, take the derivative of the function. For \(f(x) = x^2 + 2x - 3\), the derivative is \(f'(x) = 2x + 2\).
• Next, set this derivative equal to zero: \(2x + 2 = 0\). Solve for x to find the critical number. In this case, \(x = -1\).

Finding critical numbers helps in determining where a function might have changes in direction that clarify increasing or decreasing trends.

Increasing and Decreasing Intervals

Once you've identified the critical numbers, you can use them to determine on which intervals the function is increasing or decreasing. This analysis provides insights into the function's graphical behavior.

To do this:

• Place the critical numbers on a number line to divide it into intervals.
• Choose test points from these intervals and substitute them into the derivative \(f'(x)\).
• If the result is greater than zero, the function is increasing. If less, it's decreasing.

For \(f(x) = x^2 + 2x - 3\):

• Using the test point \(x = -2\) in \((-fty, -1)\), the derivative \(f'(-2) = -2\) indicates that the function is decreasing in this interval.
• Using the test point \(x = 0\) in \((-1, fty)\), the derivative \(f'(0) = 2\) shows the function is increasing in this range.

These intervals help in making sense of the overall growth pattern of the function and its application in real-world problems.

First Derivative Test

The First Derivative Test uses the derivatives to identify relative maximums, minimums, or neither at the critical points. It assesses the sign changes of \(f'(x)\) around these points.

Here's how it's done:

• Look at the sign of \(f'(x)\) before and after each critical number.
• If \(f'(x)\) changes from positive to negative, you have a relative maximum.
• Conversely, if it changes from negative to positive, there's a relative minimum.

In our example, at \(x = -1\):

• The derivative goes from negative in \((-fty, -1)\) to positive in \((-1, fty)\).
• This means the function has a relative minimum at \(x = -1\).

The First Derivative Test is a powerful tool for understanding how functions behave at specific points and tells a lot about the overall shape of the curve.