Track / Stage 5 / 357 of 400 #957 of 1964

Problem 957

AIME late

Algebra Difficulty 5.9 Prove it

1 Inequality of sequences derived from the functional inequality $\ln (1+x)>x-\frac{x^2}{2}(x>0)$

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

Prove that if $f(x)=\ln (1+x)-x+\frac{x^{2}}{2}(x>0)$ , then $f^{\prime}(x)=\frac{1}{1+x}-1+x=\frac{x^{2}}{1+x}>0$ ,

thus when $x>0$ , we have $f(x)>f(0)=0$ , i.e., $\ln (1+$ $x)>x-\frac{x^{2}}{2}$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.