18. $f ( x )$ is cubic polynomial which has local maximum at $x = - 1$. If $f ( 2 ) = 18 , f ( 1 ) = - 1$ and $f ^ { \prime } ( x )$ has local minima at $x = 0$, then\\
(A) the distance between $( - 1,2 )$ and $( \mathrm { a } , \mathrm { f } ( \mathrm { a } ) )$, where $\mathrm { x } = \mathrm { a }$ is the point of local minima is $2 \sqrt { 5 }$\\
(B) $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$\\
(C) $\mathrm { f } ( \mathrm { x } )$ has local minima at $\mathrm { x } = 1$\\
(D) the value of $\mathrm { f } ( 0 ) = 5$

Sol. (B), (C)\\
The required polynomial which satisfy the condition\\
is $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 4 } \left( 19 \mathrm { x } ^ { 3 } - 57 \mathrm { x } + 34 \right)$\\
$\mathrm { f } ( \mathrm { x } )$ has local maximum at $\mathrm { x } = - 1$ and local\\
minimum at $\mathrm { x } = 1$\\
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Hence $\mathrm { f } ( \mathrm { x } )$ is increasing for $\mathrm { x } \in [ 1,2 \sqrt { 5 } ]$.\\