question_answer

\[f\left( x \right)=\left\{ \begin{align}   & 2-\left| {{x}^{2}}+5x+6 \right|;x\ne -2 \\  & {{a}^{2}}+1\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,;x=-2 \\ \end{align} \right.\]then the range of a so that \[f\left( x \right)\] has maximum at \[x=-2\] is

A) \[\left| a \right|\ge 1\]

B) \[\left| a \right|<1\]

C) \[a>1\] 

D) \[a<1\]

Correct Answer: A

Solution :

The graph of \[y=2-\left| {{x}^{2}}+5x+6 \right|\]is drawn in the adjacent figure. Clearly/'Qc) will have maxima at

\[x=-2\]only if \[{{a}^{2}}+1\ge 2\Rightarrow \left| a \right|\ge 1\]