Continuity and Differentiability

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

Let f : (0, ) R be a function given by f(x) = cases ( 87 ) 8x 7x, & 0 < x < 2 \ a - 8, & x = 2 \ (1 + | x|) ba| x|, & 2 < x < cases where a, b Z. If f is continuous at x = 2, then a2 + b2 is equal to .

If e y = 3 -1x, then (1 - x2)y'' - xy' at x = 12 is equal to

1. 9e /2
2. 9e /6
3. 3e /2
4. 3e /6

For a, b > 0, let f(x) = cases ((a + 1)x) + b xx, & x < 0 \ 3, & x = 0 \ ax + b2x2 - axb a x x, & x > 0 cases be a continuous function at x = 0. Then ba is equal to :

1. 4
2. 5
3. 8
4. 6

If f(x) = cases x3 ( 1x ), & x 0 \ 0, & x = 0 cases, then

1. f''(0) = 0
2. f''(0) = 1
3. f'' ( 2 ) = 24 - 22
4. f'' ( 2 ) = 12 - 22