Sunday, April 9, 2017

inequality - Show that ln(x)leqx1



Show that ln(x)x1



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 ex1x then would the statement be proved?


Answer



Define for x>0



f(x)=lnxx+1f(x)=1x1=0x=1




and since f(x)=1x2<0x>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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