Let a function f(x) be defined as f ( x ) = [ ^ - 1 ( ) + x^2 ,0 … - Vidyadip

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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