Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.6 AIME Find the answer

5. Let $f(x)=x^{3}+\log _{2}\left(x+\sqrt{x^{2}+1}\right)$ . Then for any real numbers $a, b, a+b \geqslant 0$ is a ( ) for $f(a)+f(b) \geqslant 0$ .

Pick one

Solution

5.A.

Obviously, $f(x)=x^{3}+\log _{2}\left(x+\sqrt{x^{2}+1}\right)$ is an odd function and monotonically increasing. Therefore, if $a+b \geqslant 0$ , then $a \geqslant-b$ , so $f(a) \geqslant f(-b)$ , which means $f(a) \geqslant-f(b)$ . Thus, $f(a)+f(b) \geqslant 0$ .

Conversely, if $f(a)+f(b) \geqslant 0$ , then $f(a) \geqslant-f(b)=$ $f(-b)$ , which implies $a \geqslant-b$ , i.e., $a+b \geqslant 0$ .

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