Calculus: Early Transcendentals, Chapter 4, 4.6, Section 4.6, Problem 31

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

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