Let f: R R be a function defined byf(x)= l t^ 3 -3 t ; x 2 t x x^… - Vidyadip

MCQ

Let $f: R \rightarrow R$ be a function defined by$f(x)=\left\{\begin{array}{l}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\ t \leq x \\ x^{2}+2 x-6 ; 2 < x < 3 \\ {[x-3]+9 ; 3 \leq x \leq 5} \\ 2 x+1 \quad ; \quad x > 5\end{array}\right\}$ Where $[t]$ is the greatest integer less than or equal to $t$. Let $m$ be the number of points where $f$ is not differentiable and $I =\int\limits_{-2}^{2} f( x ) dx$. Then the ordered pair $( m , I )$ is equal to

A
$\left(3, \frac{27}{4}\right)$
B
$\left(3, \frac{23}{4}\right)$
C
$\left(4, \frac{27}{4}\right)$
D
$\left(4, \frac{23}{4}\right)$

✓

Answer

$\left\{\begin{array}{l} f ( x )= x ^{3}-3 x , x \leq-1 \\ 2,-1< x <2 \\ x ^{2}+2 x -6,2< x <3 \\ 9,3 \leq x <4 \\ 10,4 \leq x <5 \\ 11, x =5 \\ 2 x +1, x >5\end{array}\right.$
Clearly $f ( x )$ is not differentiable at
$x =2,3,4,5 $
$\Rightarrow m =4$
$I =\int\limits_{-2}^{-1}\left( x ^{3}-3 x \right) dx +\int\limits_{-1}^{2} 2 \cdot dx =\frac{27}{4}$

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