I taught English to Elementary school-age children in Taiwan over the summer.
Determining Concavity In Exercises $5-16,$ determine the open intervals on which the graph of the function is concave upward or concave downward. $f(x)=\frac{x^{2}+1}{x^{2}-1}$
Using the Second Derivative Test In Exercises $33-44$ , find all relative extrema of thefunction. Use the Second Derivative Test where applicable.$f(x)=x^{2 / 3}-3$
Sketching a Graph Consider a function $f$ such that $f^{\prime}$ is increasing. Sketch graphs of $f$ for (a) $f^{\prime}<0$ and (b) $f^{\prime}>0 .$
Sketching Graphs In Exercises 51 and $52,$ the graph of $f$ is shown. Graph $f, f^{\prime}$ , and $f^{\prime \prime}$ on the same set of coordinate axes. To print an enlarged copy of the graph, go to MathGraphs.com.
Think About It In Exercises $53-56,$ sketch the graph of a function $f$ having the given characteristics.$\begin{array}{l}{f(0)=f(2)=0} \\ {f^{\prime}(x)>0 \text { for } x<1} \\ {f^{\prime}(1)=0} \\ {f^{\prime}(x)<0 \text { for } x>1} \\ {f^{\prime \prime}(x)<0}\end{array}$
Think About It In Exercises $53-56,$ sketch the graph of a function $f$ having the givencharacteristics.$\begin{array}{l}{f(2)=f(4)=0} \\ {f^{\prime}(x)<0 \text { for } x<3} \\ {f^{\prime}(3) \text { does not exist. }} \\ {f^{\prime}(x)>0 \text { for } x>3} \\ {f^{\prime \prime}(x)<0, x \neq 3}\end{array}$
