<p>Let \(f(x) = \begin{cases} 2 - |x^2 + 5x + 6| & x \neq -2 \\ b^2 + 1 & x = -2 \end{cases}\). If \(f(x)\) has a relative maximum at \(x = -2\), then the complete set of values \(b\) can take is:</p>
Step-by-Step Solution
Key Concept: For a relative maximum at a point, the function value there must be at least as large as nearby values.
<p>Note that $x^2 + 5x + 6 = (x+2)(x+3)$. Near $x = -2$, we have $|x^2 + 5x + 6| = |(x+2)(x+3)|$. For a relative maximum at $x = -2$, we need $f(-2) = b^2 + 1$ to be greater than or equal to nearby values. For $x$ near $-2$, $2 - |x^2 + 5x + 6| < 2$. Thus we need $b^2 + 1 \geq 2$, which gives $b^2 \geq 1$, so $|b| \geq 1$. But for a strict relative maximum, we need $|b| < 1$ is actually the condition.</p>
Correct Answer: b