MCQ
Let a function $f(x)$ be defined as

$\begin{gathered}
  f\left( x \right) = \left[ \begin{gathered}
  {\cos ^{ - 1}}\left( \mu  \right) + {x^2},0 < x < 1 \hfill \\
  4x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,x \geqslant 1 \hfill \\ 
\end{gathered}  \right.,f\left( x \right) \hfill \\
   \hfill \\  \end{gathered}$ can have a local minimum at $x =$  $1$, if the value of $\mu$ lies in the interval

  • $\left[ { - 1,\cos 3} \right]$
  • B
    $\left( {\cos 3,1} \right]$
  • C
    $\left( {\cos 3,\cos 1} \right)$
  • D
    $\left( {\cos 3,\cos 2} \right)$

Answer

Correct option: A.
$\left[ { - 1,\cos 3} \right]$
a
$\mathop {\lim }\limits_{x \to 1}  \ge f(1)$

$ \cos ^{-1} (H)+1 \geq 4$

$ \Rightarrow  \cos ^{-1} H \geq 3 $

$ 3 \leq \cos ^{-1} H \leq \pi $

$\Rightarrow \quad-1 \leq H \leq \cos 3$

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