# `f(x) = e^x + ce^(-x)` Describe how the graph of `f`varies as `c` varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate...

`f(x) = e^x + ce^(-x)` Describe how the graph of `f`varies as `c` varies. Graph several members of the family to illustrate the trends that you discover. In particular, you should investigate how maximum and minimum points and inflection points move when `c` changes. You should also identify any transitional values of `c` at which the basic shape of the curve changes.

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To find extremums and inflection points, first find f' and f'':

`f'_c(x) = e^x-ce^(-x),`

`f''_c(x) = e^x+ce^(-x).`

1. For c<0:

f' is always positive, f increases.

f'' has one root, `x_2(c)=(ln(|c|)/2.` f'' is negative for `xltx_2(c),` f is concave downward. f'' is positive for `xgtx_2(c),` f is concave upward. `x_2(c)` is an inflection point and it moves to the left when c increases.

2. For c=0: (`f(x)=e^x` )

f' is always positive, f increases.

f'' is always positive, f is concave upward.

3. For c>0:

f' has one root, `x_1(c)=(ln(c))/2.` f' is negative for `xltx_1(c),` f devreases. f' is positive for `xgtx_1(c),` f increases. `x_1(c)` is a local minimum and it moves to the right when c increases.

f'' is always positive, f is concave upward.

So we have three types of graphs, the only transitional value for c is zero.

Please look at the graphs here: https://www.desmos.com/calculator/clarwdfywq

The green ones are for positive c's, the red ones are for negative c's (and the blue graph for c=0).