We have #f:RR->RR,f(x)=e^(x-1)#.How to demonstrate that #f(x)>x,#for any #x inRR#\{1}?

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Answer 1 · 2017-04-18T10:01:52

See below.

Calling #g(x) = f(x)-x = e^(x-1)-x#

we have

#g(1)=1-1=0#

and considering now #x = 1pm delta# we have

#g(1pmdelta)=e^(pm delta)-(pmdelta)#

or

#g_1(1+delta)=e^delta-delta > 0#
#g_2(1-delta)=e^(-delta)+delta > 0#

so

#f(x) - x > 0# for #x in RR,x ne 1#