Answer to Question #181809 in Calculus for desmond

Question #181809

5. Use the sign of the derivative to establish the inequaltiy ln(1 + x) > x − x 2 2 , ∀ x > 0


1
Expert's answer
2021-04-29T16:15:52-0400

To establish this inequality we have to made a assumption first.

f(x)=ln(1+x)x+x2f(x) = ln(1+x) -x+x^2

Hence,

f(x)=11+x1+2xf'(x ) = \dfrac{1}{1+x}-1+2x


f(x)=x+2x21+xf'(x) = \dfrac{x+2x^2}{1+x}


For x>0x>0 , We get f(x)>0f'(x) >0


Hence, we clearly say that f(x)>f(0)f(x)>f(0)


Hence, ln(1+x)x+x2>0ln(1+x)-x+x^2>0


OR


ln(1+x)>xx2ln(1+x)>x-x^2 (Proved).


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