Chapter 4.4, Problem 61E

Let y = Ae^x + Be^−x for any constants A, B.

1. (a) Sketch the graph of the function for

$\begin{array}{l}(i)A=1,B=1(\mathrm{ii})A=1,B=-1\\ (\mathrm{iii})A=2,B=1(\mathrm{iv})A=2,B=-1\\ (\mathrm{v})A=-2,B=-1(\mathrm{vi})A=-2,B=1\end{array}$

1. (b) Describe in words the general shape of the graph if A and B have the same sign. What effect does the sign of A have on the graph?
2. (c) Describe in words the general shape of the graph if A and B have different signs. What effect does the sign of A have on the graph?
3. (d) For what values of A and B does the function have a local maximum? A local minimum? Justify your answer using derivatives.