Question

Let $f: R \rightarrow R$ be a function defined by :
$f(x)= \begin{Bmatrix}\max \left\{t^{3}-3 t\right\} ; x \leq 2 \\t \leq x \\x^{2}+2 x-6 ; 2 < x < 3 \\{[x-3]+9 ; 3 \leq x \leq 5} \\2 x+1 ; x > 5\end{Bmatrix}$
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 $I =\int\limits_{-2}^{2} f( x ) d x .$ 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 1

Answer: (C)

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