Question

Let $f(x)=\begin{cases}2-\left|x^2+5 x+6\right|, & x \neq-2 \\ b^2+1, & x=-2\end{cases}$. If $f(x)$ has relative maximum at $x=-2$, then the range of the $b$, is

• A. $| b | \geq 1$
• B. $\mid$ b $\mid< 1$
• C. $b >1$
• D. b $< 1$

Answer 1

Answer: (A)

As, $f(x)$ has relative maximum at $x=-2$,
So $f(-2) \geq \underset{x \rightarrow-2}{\text{Lim}} f(x)$
$\Rightarrow b ^2+1 \geq \underset{ x \rightarrow-2}{\text{Lim}} \left(2-\left| x ^2+5 x +6\right|\right) $
$\Rightarrow b ^2 \geq 1 \Rightarrow| b | \geq 1 $