Show that ln(x)≤x−1
I'm not really sure how to show this, it's obvious if we draw a graph of it but that won't suffice here. Could we somehow use the fact that ex is the inverse? I mean, if ex−1≥x then would the statement be proved?
Answer
Define for x>0
f(x)=lnx−x+1⟹f′(x)=1x−1=0⟺x=1
and since f″(x)=−1x2<0∀x>0 , we get a maximal point.
But also
lim
Thus, the above is a global maximal point and
\forall\,x>0\;,\;\;\;f(x)\le f(1)=0
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