Demonstratie inegalitate

[elev98]

Demonstrati ca ln(1+1/x)<1/radical(x^2+x) oricare ar fi x>0

[DD]

Fie; f(x)=ln(1+1/x)-1/√(x^2+x)→f^' (x)=-((1/(1+1/x)).1/x^2+((2x+1)/(2(x^2+x)^3/2))=
-1/(x^2+x)+(2x+1)/(2.(x^2+X)^3/2=1/(x2+x)[-1+(2x+1)/(2. √(x^2+x))]>0 pentru orce x>0
....................................tabel........................................
x......0.............................................................................+inf.
f‘(x)..+inf......................................f‘(X)>0.........................0
f(x)...-inf...............................f(x)<0...................................0 de unde inealitatea este adevarata