Question

hw 16 - first derivative test section 3.2: problem 9
(1 point)
consider the function
$f(x)=\ln (2+x^{2})$.

1. $f$ is increasing on the intervals
2. $f$ is decreasing on the intervals
3. the relative maxima of $f$ occur at $x=$
4. the relative minima of $f$ occur at $x=$

notes: in the last two, your answer should be a comma separated list of $x$ values or the word
one\.
note: you can earn partial credit on this problem.
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Explanation:

Step1: Find the derivative of \(f(x)\)

Using the chain - rule, if \(y = \ln(u)\) and \(u=2 + x^{2}\), then \(y^\prime=\frac{u^\prime}{u}\).
Since \(u^\prime = 2x\) and \(u = 2 + x^{2}\), the derivative \(f^\prime(x)=\frac{2x}{2 + x^{2}}\).

Step2: Find critical points

Set \(f^\prime(x)=0\), so \(\frac{2x}{2 + x^{2}} = 0\).
Since \(2 + x^{2}>0\) for all real \(x\), then \(2x = 0\) gives \(x = 0\).

Step3: Test intervals

• For \(x<0\) (e.g., \(x=-1\)), \(f^\prime(-1)=\frac{2\times(-1)}{2+(-1)^{2}}=\frac{-2}{3}<0\).
• For \(x > 0\) (e.g., \(x = 1\)), \(f^\prime(1)=\frac{2\times1}{2+1^{2}}=\frac{2}{3}>0\).

Answer:

1. \(f\) is increasing on the interval \((0,\infty)\)
2. \(f\) is decreasing on the interval \((-\infty,0)\)
3. The relative maxima of \(f\) occur at \(x=\text{none}\)
4. The relative minima of \(f\) occur at \(x = 0\)