Continuity and Differentiability

Let f(x)= cases 13, & x 2\ b(1- x)( -2x)2, & x> 2 cases. If f is continuous at x= 2, then the value of 03b-6 |x2+2x-3 |dx is :

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

If y= -1 ( 3 x-4 x4 x+3 x )+2 -1 ( x1+ 1-x2 ), then dydx at x= 32 is equal to :

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

Let f be a real polynomial of degree n such that f(x)=f'(x)f''(x), for all x R. If f(0)=0, then 36 (f'(2)+f''(2)+ 02 f(x) dx ) is equal to:

1. 42
2. 46
3. 56
4. 66

Let f(x) = 0 ( x - x ( 2 ) (x - 1)1 + x ( 2 )(x - 1) ), x R. Consider the following two statements : (I) f(x) is discontinuous at x = 1. (II) f(x) is continuous at x = -1. Then,

1. Neither (I) nor (II) is True
2. Only (II) is True
3. Only (I) is True
4. Both (I) and (II) are True

Let , R be such that the function f(x) = cases 2 (x2 - 2 ) + 2 x, & x < 1 \ ( + 3)x + ( - ), & x 1 cases be differentiable at all x R. Then 34( + ) is equal to

1. 48
2. 84
3. 36
4. 24

Let [t] denote the greatest integer less than or equal to t. If the function f(x) = cases b2 ( 2 [ 2( x + x) x ] ), & x < 0 \ x - 12 2xx3, & x > 0 \ a, & x = 0 cases is continuous at x = 0, then a2 + b2 is equal to :

1. 12
2. 58
3. 34
4. 916

Let f(x) = cases ax2 + 2ax + 34x2 + 4x - 3 & , x - 32, 12 \ b & , x = - 32, 12 cases be continuous at x = - 32. If f f(x) = 75, then x is equal to:

1. 1.4
2. 2
3. 1
4. 0

If f(x) = cases a|x| + x2 - 2( |x|)( |x|)x & , x 0 \ b & , x = 0 cases is continuous at x = 0, then a + b is equal to :

1. 1
2. 2
3. 0
4. 4

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

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 [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

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

Let f : (- , ) - \0\ R be a differentiable function such that f'(1) = a a2 f ( 1a ). Then a a(a + 1)2 -1 ( 1a ) + a2 - 2 e a is equal to

1. 52 + 8
2. 38 + 4
3. 34 + 8
4. 32 + 4

Suppose for a differentiable function h, h(0) = 0, h(1) = 1 and h'(0) = h'(1) = 2. If g(x) = h (ex )eh(x), then g'(0) is equal to:

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

Let f(x) = x5 + 2x3 + 3x + 1, x R, and g(x) be a function such that g(f(x)) = x for all x R. Then g(7)g'(7) is equal to :

1. 42
2. 7
3. 1
4. 14

If the function f(x) = 3x + x - 3xx3, x R, is continuous at x = 0, then f(0) is equal to :

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

If y( ) = 2 + 2 3 + 4 2 + 5 + 2, then at = 2, y'' + y' + y is equal to :

1. 12
2. 1
3. 32
4. 2

Let f : [-1, 2] R be given by f(x) = 2x2 + x + [x2 ] - [x], where [t] denotes the greatest integer less than or equal to t. The number of points, where f is not continuous, is :

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

Let f(x)=x5+2ex / 4 for all x R. Consider a function g(x) such that (g f)(x)=x for all x R. Then the value of 8 g (2) is :

1. 4
2. 2
3. 16
4. 8

Let f:R be a function given by f(x)= cases 1- 2xx2, & x<0\ , & x=0,\ 1- xx, & x>0 cases where , . If f is continuous at x=0, then 2+ 2 is equal to :

1. 48
2. 6
3. 3
4. 12

If the function f(x) = cases 72x - 9x - 8x + 1 2 - 1 + x, & x 0 \ a e 2 e 3, & x = 0 cases is continuous at x = 0, then the value of a2 is equal to

1. 968
2. 1250
3. 1152
4. 746

Let f:R be defined as : f(x)= cases a-b 2xx2; & x<0\ x2+cx+2; & 0 x 1\ 2x+1; & x>1 cases If f is continuous everywhere in R and m is the number of points where f is NOT differentiable then m+a+b+c equals :

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

Let f(x)= |2 x2+5|x|-3 |, x R. If m and n denote the number of points where f is not continuous and not differentiable respectively, then m+n is equal to :

1. 5
2. 3
3. 2
4. 0

Let g(x) be a linear function and f(x)= cases g(x) & , x 0 \ ( 1+x2+x ) 1x & , x>0 cases, is continuous at x=0. If f (1)=f(-1), then the value g(3) is

1. e ( 49 )-1
2. 13 e ( 49 e1/3 )
3. e ( 49 e1/3 )
4. 13 e ( 49 )+1

Consider the function f:(0, ) R defined by f(x)=e- | ex |. If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then m+n is

1. 0
2. 1
3. 2
4. 3