Continuity and Differentiability

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

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