Problem 35 - Calculus of a Single Variable

Question

Applying the First Derivative Test In Exercises \(17-40\) , (a) find the critical numbers of \(f\) (if any), (b) find the open interval(s) on which the function is increasing or decreasing, (c) apply the First Derivative Test to identify all relative extrema, and (d) use a graphing utility to confirm your results. $$ f(x)=\frac{x^{2}}{x^{2}-9} $$

Step-by-Step Solution

Answer

The function has critical numbers at x = 0, -3, and 3. It is decreasing on the intervals (-∞, -3) and (0, 3), and increasing on the intervals (-3, 0) and (3, +∞). The function has a local maximum at x = 0 and no local minimum.

Step 1: Find the Derivative (f'(x))

Firstly, find the derivative of the function using the quotient rule: \( f'(x) = \frac{((x^{2})'*(x^{2}-9)-(x^{2})*(x^{2}-9)')}{(x^{2}-9)^{2}} = \frac{2x * (x^{2} - 9) - x^{2} * 2x}{(x^{2}-9)^{2}} = \frac{2x^{3} - 18x - 2x^{3}}{(x^{2}-9)^{2}} = \frac{- 18x}{(x^{2} - 9)^{2}} \)

Step 2: Find the Critical Numbers

Critical numbers occur when the derivative is zero or undefined. The derivative is zero when the numerator is zero (i.e., -18x= 0), and undefined when the denominator is zero (i.e., x^{2}- 9 = 0). Solving these equations, x = 0, -3, and 3 are the critical numbers.

Step 3: Determine Increasing and Decreasing Intervals

Create a number line using the critical numbers. Choose a sample point from each interval and evaluate the derivative at that point to determine whether the function is increasing or decreasing. If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing. Here, the function is decreasing on the intervals (-∞, -3) and (0, 3), and increasing on the intervals (-3, 0) and (3, +∞).

Step 4: Apply the First Derivative Test

Using the intervals established in Step 3 and the first derivative test, identify where the function transitions from decreasing to increasing (local minimum) or increasing to decreasing (local maximum). Here, the function has a local maximum at x=0 and no local minimum.

Step 5: Graph the Function

Graphing the function will visually confirm these results. The critical numbers should appear as local extrema, and the function should be increasing or decreasing on the intervals as determined in Step 3.

Key Concepts

Critical Numbers

A core concept in calculus is the idea of critical numbers, which are pivotal values where a function's graph changes its course. To uncover these key players, we investigate the function's derivative, searching for points where it is either zero or doesn't exist.

In the context of the exercise, we applied this notion to the function \( f(x)=\frac{x^{2}}{x^{2}-9} \). The derivative, \( f'(x) \), was calculated and set equal to zero, revealing critical numbers at \( x = 0 \); and where the derivative is undefined, leading us to \( x = -3, 3 \). These are the precise points at which the function's graph halts its present course and potentially redirects—either peaking, dipping, or experiencing a kink in its curve.

Increasing and Decreasing Intervals

Armed with the vital critical numbers, our next mission is to chart the paths our function travels—where it ascends and descends. We sketch a number line, planting the critical numbers as markers, then select test points from the intervals they create.

A derivative greater than zero implies the function is climbing—increasing. A negative derivative indicates fading brightness—a decreasing function. Applying this to our function, investigators revealed that the graph descends into the valley of decrescendo between \( (-\infty, -3) \) and \( (0, 3) \), and ascends the slopes of crescendo from \( (-3, 0) \) to \( (3, +\infty) \). This rhythmic alternation maps out the undulations of our function's journey.

Local Extrema

Weaving together the threads of increasing and decreasing intervals, we're now poised to uncover the local extrema—those spellbinding peaks and troughs on the function's landscape. Grasping the First Derivative Test as our compass, we discern these notable landmarks on the function's terrain where it shifts gears.

For the function \( f(x) \), we recognized a local maximum at \( x = 0 \), where the function switches from increasing prelude to decreasing sequel. No local minimum was spotted, as the function did not transition from a decline to an ascent at any of the critical points.

Graphing Functions

A powerful way to visualize a function's story is through graphing. This graphical representation serves as both confirmation and exploration—a mirror to our mathematical deductions. Utilizing a graphing utility can cast light on the narrative woven by our calculations.

Graphing \( f(x) \), we expect to see the function sulking downwards to the left and right of \( x=-3 \) and \( x=3 \), taking a gallant ascent in between these critical numbers, and at \( x=0 \), perched at a majestic peak—the local maximum, as deciphered from our analysis. A glance at the graph authoritatively confirms our calculated intervals and extrema—et voilà—the numbers spring to life, etched into the canvas of the coordinate plane.