[derivative] prove inequality in x < x < e ^ x, x > 0 by monotonicity It is proved that in x < x < e ^ x,

f(x)=lnx,g(x)=e^x
Do parallel to the line y = x and f (x) = LNX, G (x) = e ^ x respectively
Tangent line L1: y = x + m, L2: y = x + n
F '(x) = 1 / x, Let f' (x) = 1, that is, 1 / x = 1 = = > x = 1, f (1) = 0
h'(x)=e^x,e^x=1==>x=0,y=1
l1:y=x-1,l2:y=x+1
u(x)= f(x)-x+1=lnx-x+1,u'(x)=1/x-1=(1-x)/x
00, V (x) is an increasing function
v(x)>v(0)=0∴e^x-x-1>0
==> e^x>x+1
And X-1 < xlnx ≤ X / E

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