Problem 24 - Essential Calculus

Question

Find the critical numbers of \(f\) (if any). Find the open intervals on which the function is increasing or decreasing and locate all relative extrema. Use a graphing utility to confirm your results. $$ f(x)=(x+2)^{2}(x-1) $$

Step-by-Step Solution

Answer

The critical numbers of the function are -2 and 1. The function is decreasing on the intervals \((-∞, -2)\) and \((1, ∞)\), and increasing on the interval \((-2, 1)\). There is a relative minimum at \(x=-2\) and a relative maximum at \(x=1\). The results are confirmed with a graphing utility.

Step 1: Function Differentiation

Begin by finding the derivative of the function, \(f'(x)\). Use the product rule, which states that the derivative of two functions multiplied together is the derivative of the first times the second plus the first times the derivative of the second. Apply the power rule where necessary, which states that the derivative of \(x^n\) is \(nx^{n-1}\). The derivative of \(f(x)=(x+2)^{2}(x-1)\) is \(f'(x)=2(x+2)(x-1)+(x+2)^{2}\).

Step 2: Locating Critical Numbers

The next step is to find the critical numbers. These are points where \(f'(x)=0\) or where \(f'(x)\) is undefined. However, since \(f'(x)\) is a polynomial, it is defined for all \(x\). Setting \(f'(x)\) equal to zero and solving gives \(x=-2\) and \(x=1\) as the critical numbers.

Step 3: Determine Increasing and Decreasing Intervals

Use the first derivative test to determine where the function is increasing or decreasing. Test points in each interval \((-∞, -2)\), \((-2, 1)\), \((1, ∞)\) in the derivative expression. If the derivative at a test point is positive, then the function is increasing in that interval. If it is negative, then it's decreasing. Through this testing we find that the function is decreasing on \((-∞, -2)\) and \((1, ∞)\), and increasing on \((-2, 1)\).

Step 4: Finding Relative Extrema

Since the function changes from decreasing to increasing at \(x=-2\), there is a relative minimum at \(x=-2\). Similarly, as it changes from increasing to decreasing at \(x=1\), there is a relative maximum at \(x=1\).

Step 5: Confirmation with Graphing Utility

To confirm the results, graph the function \(f(x)=(x+2)^{2}(x-1)\) using a graphing utility. The graph indeed confirms a relative minimum at \(x=-2\) and a relative maximum at \(x=1\). The intervals of increase and decrease also align with our results.

Key Concepts

Derivative

The concept of the derivative is central to calculus, and it's all about understanding how a function changes. The derivative of a function represents the rate at which the function's value changes as the input changes. In mathematical terms, for a given function \( f(x) \), the derivative, denoted \( f'(x) \), is defined as:

• The limit as \( h \) approaches zero of \( \frac{f(x+h) - f(x)}{h} \).

For practical calculations, there are several rules, like the product rule and power rule, which simplify finding derivatives. In this exercise, we used them to differentiate \( f(x)=(x+2)^2(x-1) \). The product rule helped us deal with the multiplication of two functions, resulting in the derivative \( f'(x) = 2(x+2)(x-1) + (x+2)^2 \). Understanding these rules is essential for finding critical numbers, intervals of increase or decrease, and extrema.

Critical Numbers

Critical numbers of a function are specific values of \( x \) where the derivative \( f'(x) \) is zero or undefined. These numbers are crucial because they are potential locations for relative extrema—points where the function reaches a local maximum or minimum. To find these, we:

• Set the derivative equal to zero: \( f'(x) = 0 \).
• Solve the equation for \( x \).

In our problem, solving \( f'(x) = 0 \) led us to the critical numbers \( x = -2 \) and \( x = 1 \). Since the derivative is a polynomial, it's defined for all values of \( x \), so there are no points where \( f'(x) \) is undefined.

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we use the first derivative test. This test involves:

• Identifying intervals between and beyond the critical numbers.
• Selecting test points in each interval and calculating the sign of \( f'(x) \) at these points.
• The sign of \( f'(x) \) shows if the function is increasing or decreasing:
• If \( f'(x) > 0 \), the function is increasing.
• If \( f'(x) < 0 \), the function is decreasing.

In our exercise:

• The function is decreasing on \((-∞, -2)\) and \((1, ∞)\) since \( f'(x) < 0 \) in these intervals.
• It is increasing on \((-2, 1)\) as \( f'(x) > 0 \) here.

This analysis helps identify where changes in the direction of the function's slope occur.

Relative Extrema

Relative extrema refer to the local maximums and minimums of a function within a specific interval. To find these points, we see where the function goes from increasing to decreasing (for a maximum) or decreasing to increasing (for a minimum). This change indicates a peak or trough in the graph:

• If \( f'(x) \) changes from negative to positive at a critical number, it's a relative minimum.
• If \( f'(x) \) changes from positive to negative, it's a relative maximum.

In this exercise:

• A relative minimum occurs at \( x = -2 \) as the slope changes from negative to positive.
• A relative maximum occurs at \( x = 1 \) as the slope changes from positive to negative.

Using a graphing utility can visually confirm these points, helping verify calculations.