Question One

Consider the function $f(x)=\frac{1}{1+x^2}$

$($a$)$ Find the first derivative. Clearly show all the steps and state what differentiation rules were used.

$($b$)$ Do a sign analysis on the first derivative. Clearly show all the steps and explain the process in words.

$($c$)$ State the interval$($s$)$ where the function $f$ is increasing and the interval$($s$)$ where the function $f$ is decreasing. State the location and type of any local extrema.

$($d$)$ Find the second derivative. Clearly show all the steps and state what differentiation rules were used.

$($e$)$ Do a sign analysis on the second derivative. Clearly show all the steps and explain the process in words.

$($f$)$ State the interval$($s$)$ where the function $f$ is concave up and the interval$($s$)$ where the function $f$ is concave down. State the location of any points of inflection.

$($g$)$ Draw the graph. Label the information found above on the graph.