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Please answer the following questions about the function $$f(x)=\frac{3 x^{2}}{x^{2}-16}$$
Instructions: - If you are asked for a function, enter a function. - If you are asked to find $x$ - or $y$-values, enter either a number or a list of numbers separated by commas. If there are no solutions, enter None. - If you are asked to find intervals of increase/decrease/concavity, please separate the lists with commas and NOT the union symbol. - For the domain and range, use interval notation and NOT commas. - If you are asked to find a limit, enter DNE if the limit does not exist. (a) Calculate the first derivative of $f$. Find the critical numbers of $f$, where it is increasing and decreasing, and its local extrema. $$f^{\prime}(x)=\frac{-96 x}{\left(x^{2}-16\right)^{2}}$$
Critical numbers $x=$ $\square$ List of the intervals where $f(x)$ is increasing (separated by commas and NOT the union symbol) $(-\infty,-4),(-4,0)$ List of the intervals where $f(x)$ is decreasing (separated by commas and NOT the union symbol) $(0,4),(4, \infty)$ Local maxima $x=$ $\square$ Local minima $x=$ $\square$ (b) Find the following left- and right-hand limits at the vertical asymptote $x=-4$. $$\lim _{x \rightarrow-4^{-}} \frac{3 x^{2}}{x^{2}-16}=+ \text { infinity } \leqslant \quad \lim _{x \rightarrow-4^{+}} \frac{3 x^{2}}{x^{2}-16}=- \text { infinity }$$
Find the following left- and right-hand limits at the vertical asymptote $x=4$. $$\lim _{x \rightarrow 4^{-}} \frac{3 x^{2}}{x^{2}-16}=- \text { Infinity } \rightarrow \quad \lim _{x \rightarrow 4^{+}} \frac{3 x^{2}}{x^{2}-16}=+ \text { infinity } \leqslant$$
Find the following limits at infinity to determine any horizontal asymptotes. $$\lim _{x \rightarrow-\infty} \frac{3 x^{2}}{x^{2}-16}=3$$ (c) Calculate the second derivative of $f$. Find where $f$ is concave up, concave down, and has inflection points. $$f^{\prime \prime}(x)=\frac{96\left(3 x^{2}+16\right)}{\left(x^{2}-16\right)^{3}}$$
List of the intervals where $f(x)$ is concave up (separated by commas and NOT the union symbol) $(-\infty,-4),(4, \infty)$ List of the intervals where $f(x)$ is concave down (separated by commas and NOT the union symbol) $(-4,4)$ Inflection points $x=$ None (d) The function $f$ is even $\leqslant$ because $f(-x)=f(x) \quad y$ for all $x$ in the domain of $f$, and therefore its graph is symmetric about the $y$-axis (e) Answer the following questions about the function $f$ and its graph.
The domain of $f$ is the set (in interval notation and NOT using commas) $(-\infty,-4) \cup(-4,4) \cup(4, \infty)$ The range of $f$ is the set (in interval notation and NOT using commas) 1 $y$-intercept 0 $x$-intercepts 0