14. If $f ( x ) = \min \left\{ 1 , x ^ { 2 } , x ^ { 3 } \right\}$, then\\
(A) $\mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$\\
(B) $\mathrm { f } ^ { \prime } ( \mathrm { x } ) > 0 , \forall \mathrm { x } > 1$\\
(C) $\mathrm { f } ( \mathrm { x } )$ is not differentiable but continuous $\forall \mathrm { x } \in \mathrm { R }$\\
(D) $f ( x )$ is not differentiable for two values of $x$

Sol. (A), (C)\\
$\mathrm { f } ( \mathrm { x } ) = \min . \left\{ 1 , \mathrm { x } ^ { 2 } , \mathrm { x } ^ { 3 } \right\}$\\
$\Rightarrow \mathrm { f } ( \mathrm { x } ) = \begin{cases} \mathrm { x } ^ { 3 } , & \mathrm { x } \leq 1 \\ 1 , & \mathrm { x } > 1 \end{cases}$\\
$\Rightarrow \mathrm { f } ( \mathrm { x } )$ is continuous $\forall \mathrm { x } \in \mathrm { R }$ and non-differentiable at $\mathrm { x } = 1$.\\
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