Functions
Functions
Allen Star Batch
Grade 12
Question:
Suppose $f(x) = x^3 + \log_2\left(x + \sqrt{x^2 + 1}\right)$. For any $a, b \in \mathbb{R}$ to satisfy $f(a) + f(b) \geq 0$, the condition $a + b \geq 0$ is:
Necessary and sufficient
Necessary but not sufficient
Not necessary but sufficient
Neither necessary nor sufficient
Step-by-Step Solution
Key Concept: Verify that $f(x) = x^3 + \log_2(x + \sqrt{x^2 + 1})$ is an odd function ($f(-x) = -f(x)$) and strictly increasing. Then $f(a) + f(b) \geq 0 \iff f(a) \geq -f(b) = f(-b) \iff a \geq -b \iff a + b \geq 0$ by monotonicity.
Notice that $f(x)$ is odd and increasing. If $a+b \geq 0$ and $a \geq -b$, then $f(a) \geq f(-b)$ or $f(a) \geq -f(b)$, so $f(a) + f(b) \geq 0$, and these inequalities can be traced back through the monotonicity.
Correct Answer: 1