Question

$f\left(x\right)=\{\begin{array}{c}2-|x^2+5x+6|,x\ne -2\\ a^2+1,x=-2\end{array}$. then the range of $a$, so that $f\left(x\right)$ has maxima at $x=-2$, is

• A. $|a|\ge 1$
• B. $|a|<1$
• C. $a>1$
• D. $a<1$

Answer 1

The correct option is A $|a|\ge 1$

According to question,

$f\left(x\right)$ has maximum at x = -2

$\begin{array}{c}\Rightarrow f(-2)\ge {lim}_{x\to -2}2-\left|x^2+5x+6\right|\\ \Rightarrow a^2+1\ge 2-\left|4-10+6\right|\\ \Rightarrow a^2+1\ge 2-0\\ \Rightarrow a^2\ge 1\\ \therefore \left|a\right|\ge 1\end{array}$