Let f: R arrow R be a function defined by: f(x)= beginBmatrix max t3-3 t ; x ≤ 2 t ≤ x x2+2 x-6 ; 2

Question

Let f: R arrow R be a function defined by: f(x)= beginBmatrix max t3-3 t ; x ≤ 2 t ≤ x x2+2 x-6 ; 2 < x < 3 [x-3]+9 ; 3 ≤ x ≤ 5 2 x+1 ; x > 5 endBmatrix 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 =∫ limits-22 f( x ) d x . Then the ordered pair ( m , I ) is equal to : (A) (3, (27/4)) (B) (3, (23/4)) (C) (4, (27/4)) (D) (4, (23/4))

Tardigrade Admin · Accepted Answer

begincasesf(x)=x3-3 x, x ≤-1 2,-1 < x < 2 x2+2 x-6,2 < x < 3 9,3 ≤ x < 4 10,4 ≤ x < 5 11, x=5 2 x+1, x > 5 endcases Clearly f ( x ) is not differentiable at x =2,3,4,5 ⇒ m =4 I=∫ limits-2-1(x3-3 x) d x+∫ limits-12 2 ⋅ d x=(27/4)

$(3, \frac{27}{4})$

$(3, \frac{23}{4})$

$(4, \frac{27}{4})$

$(4, \frac{23}{4})$

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

(3,427​)

(3,423​)

(4,427​)

(4,423​)

⎩⎨⎧​f(x)=x3−3x,x≤−12,−1<x<2x2+2x−6,2<x<39,3≤x<410,4≤x<511,x=52x+1,x>5​ Clearly f(x) is not differentiable at x=2,3,4,5⇒m=4 I=−2∫−1​(x3−3x)dx+−1∫2​2⋅dx=427​

$(3, \frac{27}{4})$

$(3, \frac{23}{4})$

$(4, \frac{27}{4})$

$(4, \frac{23}{4})$

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