Continuity and Differentiability

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 [t] denote the greatest integer less than or equal to t. Let f : [0, ) R be a function defined by f(x) = [ x2 + 3 ] - [ x]. Let S be the set of all points in the interval [0, 8] at which f is not continuous. Then a S a is equal to .

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