Continuity and Differentiability

Let [ ] denote the greatest integer function, and let f(x) = \ 2x, x2 \. Let S = \x (-2, 2) : the function g(x) = |x| [x2 ] is discontinuous at x\. Then x S f(x) equals

1. 2 - 2
2. 2 6 - 3 2
3. 1 - 2
4. 6 - 2 2

The number of points of discontinuity of the function f(x) = [ x22 ] - [ x], x [0, 4], where [ ] denotes the greatest integer function, is .

If the function f(x) = ( x) - ( x) x - x is continuous at x = 0, then f(0) is equal to .

Let m and n be the number of points at which the function f(x) = \x, x3, x5, , x21 \, x R, is not differentiable and not continuous, respectively. Then m + n is equal to .

Let f(x) = cases (1 + ax)1/x, & x < 0 \ 1 + b, & x = 0 \ (x+4)1/2 - 2(x+c)1/3 - 2, & x > 0 cases be continuous at x = 0. Then ea bc is equal to:

1. 64
2. 48
3. 36
4. 72

Let the function f(x)= (x2-1 ) |x2-ax+2 |+ |x| be not differentiable at the two points x= =2 and x= . Then the distance of the point ( , ) from the line 12x+5y+10=0 is equal to :

1. 5
2. 2
3. 4
4. 3

Let f(x) = cases 3x, & x < 0 \ \1 + x + [x], x + 2[x]\, & 0 x 2 \ 5, & x > 2 cases where [.] denotes greatest integer function. If and are the number of points, where f is not continuous and is not differentiable, respectively, then + equals .

Let [x] denote the greatest integer function, and let m and n respectively be the numbers of the points, where the function f(x)=[x]+|x-2|, -2<x<3, is not continuous and not differentiable. Then m+n is equal to :

1. 6
2. 9
3. 8
4. 7

If the function f(x)= cases 2x \ (k1+1)x+ (k2-1)x \, & x<0\\[2mm] 4, & x=0\\[2mm] 2x e ( 2+k1x2+k2x ), & x>0 cases is continuous at x=0, then k12+k22 is equal to :

1. 5
2. 10
3. 20
4. 8

Let f(x) = ax3 + bx2 + cx + 41 be such that f(1) = 40, f'(1) = 2 and f''(1) = 4. Then a2 + b2 + c2 is equal to:

1. 54
2. 51
3. 73
4. 62