China Mathematical Competition

Question

Suppose f(x) = x^3 + _2(x + x^2+1). For any a, b R, to satisfy f(a) + f(b) 0, the condition a + b 0 is ( ). (A) necessary and sufficient (B) not necessary but sufficient (C) necessary but not sufficient (D) neither necessary nor sufficient

Accepted Answer

Obviously f(x) = x^3 + _2(x + x^2+1) is an odd function and is monotonically increasing. So, if a+b 0, i.e. a -b, we get f(a) f(-b), f(a) -f(b), and that means f(a)+f(b) 0. On the other hand, if f(a) + f(b) 0, then f(a) -f(b) = f(-b). So a -b, a+b 0. Answer: A.