Summarize the pertinent information obtained by applying the graphing strategy and sketch the graph of $$ f(x)=\ln \left(x^{2}+64\right) $$. Summarize the pertinent information obtained by analyzing $$ f^{\prime}(x) $$. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. $$ f(x) $$ is increasing on $$ \square $$ and decreasing on B. $$ f(x) $$ is increasing on $$ f(x) $$ is decreasing on $$ \square $$ $$ f(x) $$ has a local Summarize the pertinent information obtained by analyzing $$ f^{\prime \prime}(x) $$. Select the correct choice below and fill in the answer box(es) to complete your choice. (Type your answer in interval notation. Use a comma to separate answers as needed.) A. $$ f(x) $$ is concave upward on $$ \square $$ and concave downward on $$ \square $$. B. $$ f(x) $$ is concave upward on C. $$ f(x) $$ is concave downward on

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**Domain and Symmetry** The function is $$ f(x)=\ln(x^2+64) $$ Since $$x^2+64>0$$ for all $$x$$, the domain is $$ (-\infty,\infty) $$ Also, because $$x^2$$ is even, $$f(x)$$ is an even function (symmetric about the $$y$$-axis). **Intercept** At $$x=0$$, $$ f(0)=\ln(0^2+64)=\ln(64) $$ so the graph passes through $$(0,\ln(64))$$. **Analysis of $$f'(x)$$ (Monotonicity and Local Extreme)** Differentiate $$f(x)$$ using the chain rule. Let $$u=x^2+64$$; then $$f(x)=\ln(u)$$ and $$ f'(x)=\frac{u'}{u}=\frac{2x}{x^2+64}. $$ *Critical Point:* Set $$f'(x)=0$$: $$ \frac{2x}{x^2+64}=0\quad\Longrightarrow\quad x=0. $$ *Sign of $$f'(x)$$:* - For $$x<0$$: $$2x<0$$ (while $$x^2+64>0$$), so $$f'(x)<0$$. - For $$x>0$$: $$2x>0$$, so $$f'(x)>0$$. Thus, the function is decreasing on $$ (-\infty,0) $$ and increasing on $$ (0,\infty). $$ There is a local minimum at $$x=0$$, with $$ f(0)=\ln(64). $$ **Analysis of $$f''(x)$$ (Concavity and Inflection Points)** Differentiate $$f'(x)=\frac{2x}{x^2+64}$$ using the quotient rule: $$ f''(x)=\frac{(2)(x^2+64)-2x\cdot(2x)}{(x^2+64)^2}=\frac{2(x^2+64)-4x^2}{(x^2+64)^2}=\frac{128-2x^2}{(x^2+64)^2}. $$ This can be factored as $$ f''(x)=\frac{2(64-x^2)}{(x^2+64)^2}. $$ The denominator is always positive, so the sign of $$f''(x)$$ depends on the numerator $$2(64-x^2)$$. Set $$64-x^2>0$$: $$ 64-x^2>0\quad\Longrightarrow\quad x^2<64\quad\Longrightarrow\quad -80$$ when $$-88$$ (concave downward). Inflection points occur at $$x=-8$$ and $$x=8$$. Evaluating the function at these points: \[ f(-8)=\ln((-8)^2+64)=\ln(64+64)=\ln(128), $$ $$ f(8)=\ln(8^2+64)=\ln(64+64)=\ln(128). $$ **Summary of Answers** For the first choice (monotonicity and local extreme): - $$f(x)$$ is increasing on $$\boxed{(0,\infty)}$$ and decreasing on $$\boxed{(-\infty,0)}$$. - $$f(x)$$ has a local minimum at $$x=0$$ with value $$\ln(64)$$. For the second choice (concavity): - $$f(x)$$ is concave upward on $$\boxed{(-8,8)}$$ and concave downward on $$\boxed{(-\infty,-8),\ (8,\infty)}$$. **Summary of Analysis for $$ f(x) = \ln(x^2 + 64) $$:** 1. **Domain and Symmetry:** - *Domain:* All real numbers ($$ (-\infty, \infty) $$) - *Symmetry:* Even function (symmetric about the y-axis) 2. **Intercept:** - Passes through $$(0, \ln(64))$$ 3. **Monotonicity and Local Extreme:** - *Increasing:* $$ (0, \infty) $$ - *Decreasing:* $$ (-\infty, 0) $$ - *Local Minimum:* At $$ x = 0 $$ with value $$ \ln(64) $$ 4. **Concavity:** - *Concave Upward:* $$ (-8, 8) $$ - *Concave Downward:* $$ (-\infty, -8) $$ and $$ (8, \infty) $$ - *Inflection Points:* At $$ x = -8 $$ and $$ x = 8 $$ **Choices:** - A. $$ f(x) $$ is increasing on $$ (0, \infty) $$ and decreasing on $$ (-\infty, 0) $$ - B. $$ f(x) $$ is concave upward on $$ (-8, 8) $$ and concave downward on $$ (-\infty, -8) $$ and $$ (8, \infty) $$

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