91Ó°ÊÓ

A function \(f\) with domain either \(I=(-\infty, \infty)\) or \(I=(0, \infty)\) is given. Sketch the graph of \(f\). (The set \(C\) of critical points of \(f\) and the set \(I\) of inflection points of \(f\) are provided in some cases.) Use l'Hôpital's Rule to determine the horizontal asymptote of the graph. If \(I=(0, \infty),\) use l'Hôpital's Rule to determine \(\lim _{x \rightarrow 0^{+}} f(x)\). \(f(x)=\ln (x) /\left(1 / x+e^{x-1}\right) ; \quad I=(0, \infty) ; \quad C=\\{0.730 \ldots\\} ; I=\) \(\\{1.646 \ldots\\}\)

In each of Exercises 82-89, use the first derivative to determine the intervals on which the function is increasing and on which the function is decreasing. $$ f(x)=2 x^{5}-1.2 x^{4}-13.7 x+5.2 $$

For each given function \(f,\) graph the function \(S(x)=\) signum \(\left(f^{\prime}(x)\right)\) for \(x\) in the given interval \(I .\) Use the graph of \(S\) to determine and classify the local extrema of \(f\). a. \(f(x)=x^{5}-12 x^{4}+55 x^{3}-120 x^{2}+124 x-48\) \(I=[0.8,4.1]\) b. \(f(x)=4 x^{4}-24 x^{3}+51 x^{2}-44 x+12, \quad I=[0.3,2.6]\) c. \(f(x)=x^{4}-x^{3}-7 x^{2}+x+6, \quad I=[-2.4,3.2]\)