Problem 38 - 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. $$ f(x)=\frac{x^{2}-3 x-4}{x-2} $$

Step-by-Step Solution

Answer

The function has one critical number at x = 2 but it has no relative extrema. The function is decreasing for all x.

Step 1: Derive the Function

Deriving the function using quotient rule which states that for any two differentiable functions h(x) and g(x), the derivative of the quotient h(x) / g(x) is given by \[ f'(x) = \frac{g(x)h'(x) - h(x)g'(x)}{(g(x))^2} \]. Therefore, the derivative of the function will be: \[ f'(x) = \frac{(x-2)(2x - 3) - (x^{2} - 3x - 4)}{(x-2)^2} = \frac{-x-2}{(x-2)^2}\. \]

Step 2: Determine the Critical Numbers

The critical numbers are given by the values of x for which f'(x) = 0 or f'(x) is undefined. Solving for f'(x) = 0 generates no solution, but f'(x) is undefined when x = 2. So, 2 is a critical number.

Step 3: Test Intervals

To determine where the function is increasing or decreasing, it is essential to test intervals. The intervals are determined by the critical number, which are (-∞, 2) and (2, ∞). Choose test points within these intervals, such as 1 in (-∞, 2) and 3 in (2, ∞) and substitute these into the derivative. For x=1, f'(1) = -3 which is less than 0, indicating the function is decreasing in the interval (-∞, 2). For x=3, f'(3) = -2/3 which is less than 0, so the function is decreasing in the interval (2, ∞).

Step 4: Identify the Relative Extrema

Since the function is decreasing before and after x = 2, there are no relative maxima or minima.

Step 5: Verify with a Graphing Utility

Use a graphing utility to verify these findings. The graph of the function verifies that it is always decreasing and so there are no relative maxima or minima.

Key Concepts

Graphing Utility Analysis

Graphing utilities are powerful tools that assist in visualizing the functions and confirming analytical results. After you've performed an analysis on paper and calculated where the function is increasing, decreasing, or located any relative extrema, a graphing utility can help verify these findings. It provides a visual representation, which is particularly useful in complex functions where the behavior isn't immediately evident. In our example, graphing the function confirms that it is always decreasing, as indicated by the derivative's analysis. Hence, we can conclude that no relative extrema exist, and our previous work is substantiated by the graph. Utilizing such tools not only provides confirmation but also enhances comprehension of the function's overall behavior.