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the example f is concave-up for all real x as may be seen by, in the leftmost c range, induction. |
f''(x)=2e^x / (1+2e^x)^2 > 0 in the leftmost x-range, and by induction for x>0. |
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<math>f''\ge 0</math> everywhere (i.e. <math>f(x)</math> is concave-up,
and <math>f'(x)</math> increasing,
for all real <math>x
is to take <math>A=\tfrac12</math> and <math>g(x)=x+\tfrac12</math>, giving
<math display=block> f (x) =
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