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1.Relation and Function
normal
Let $f(x) = \left\{ {\begin{array}{*{20}{c}}
{\,{x^3} - {x^2} + 10x - 5\,\,,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x \le 1\,\,\,\,\,\,\,\,\,\,\,\,}\\
{ - 2x + {{\log }_2}({b^2} - 2),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,x\, > 1\,\,\,\,\,\,\,\,\,\,\,\,}
\end{array}} \right.$ the set of values of $b$ for which $f(x)$ has greatest value at $x = 1$ is given by
A
$1 \le b \le 2$
B
$b = \{ 1,2\} $
C
$b \in ( - \infty , - 1)$
D
$\left[ { - \sqrt {130} , - \sqrt 2 } \right) \cup \left( {\sqrt 2 ,\sqrt {130} } \right]$
Solution
$f\left(1^{-}\right) \leq f(1)$ and $f\left(1^{+}\right) \leq f(1)$
$-2+\log _{2}\left(b^{2}-2\right) \leq 5$
$0 < {b^2} – 2 \le 128\quad 2 < {b^2} \le 130$
Standard 12
Mathematics
