Problem 35 - 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. $$ f(x)=\frac{x^{2}}{x^{2}-9} $$

Step-by-Step Solution

Answer

The critical numbers of \(f(x)\) are \(3\) and \(-3\), and the function \(f(x)\) is increasing for \(x < -3\), decreasing for \(-3 < x < 3\), and again increasing for \(x > 3\). The function has a relative maximum at \(x = -3\) and a relative minimum at \(x = 3\).

Step 1: Finding Derivative of The Function

To find the derivative of function \(f(x)\), we can use quotient rule which states that the derivative of \(\frac{u}{v}\) is \(\frac{vu' - uv'}{v^{2}}\), where \(u'\) and \(v'\) are the derivatives of \(u\) and \(v\) respectively.\nNow, let \(u=x^{2}\) and \(v=x^{2}-9\). We find \(u'\) by differentiating \(x^{2}\) to get \(2x\) and \(v'\) by differentiating \(x^{2}-9\) to get \(2x\).\nSubstitute these into the quotient rule, we get: \n\(f'(x)=\frac{(x^{2}-9)*2x - x^{2}*2x}{(x^{2}-9)^{2}} = \frac{-18x}{(x^{2}-9)^{2}}\)

Step 2: Finding the Critical Points

The critical numbers of \(f(x)\) are the solutions to the function \(f'(x) = 0\) and places where \(f'(x)\) doesn't exist. Here \(f'(x)\) is undefined when the denominator equals zero, which is when \(x=3\) or \(x=-3\). Setting the numerator to zero, we find that there are no solutions because -18x is never 0. So, the critical points are \(x = 3\) and \(x = -3\). Note that the function is undefined for these two values of \(x\).

Step 3: Analyze Increase and Decrease

To find the intervals for which the function is increasing or decreasing, we select a test point from each of the intervals determined by the critical points and substitute it into the derivative function.\nThe intervals are \(-\infty, -3\), \(-3, 3\) and \(3, +\infty\). Choose -4, 0, and 4 as test points. Evaluating \(f'(-4)\), \(f'(0)\), and \(f'(4)\) we get positive, undefined and negative values respectively. So, \(f(x)\) is increasing when \(x< -3\), and decreasing when \(x>3\).

Step 4: Finding Relative Extrema

From above intervals, we notice that the function changes from increasing to undefined and then to decreasing. This indicates that \(x = -3\) is a relative maximum. Similarly, \(f(x)\) changes from decreasing to undefined and then to increasing, which indicates that \(x = 3\) is a relative minimum.

Step 5: Graphing the Function

Using a graphing utility, we can sketch a graph of the function to confirm our results. The graph should indicate that the function increases from \(-\infty\) to \(-3\), decreases from \(-3\) to \(3\), and increases again from \(3\) onwards. Moreover, the function has relative maximum at \(x = -3\) and relative minimum at \(x = 3\).