Continuity and Differentiability

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

If y= ( x+1) (x2- x )x x+x+ x+ 115 (3 2x-5 ) 3x, then 96y' ( 6 ) is equal to :

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

If the function f(x)= cases 1|x|, & |x| 2 \ ax2+2b, & |x|<2 cases is differentiable on R, then 48(a+b) is equal to .

Let a and b be real constants such that the function f defined by f(x)= casesx2+3x+a, & x 1\ bx+2, & x>1 cases be differentiable on R. Then, the value of -22f(x) dx equals

1. 21
2. 19/6
3. 17
4. 15/6

Let y= e ( 1-x21+x2 ), -1<x<1. Then at x= 12, the value of 225 (y'-y'' ) is equal to

1. 732
2. 736
3. 742
4. 746

Consider the function. f(x)= cases a (7x-12-x2 )b |x2-7x+12 |, & x<3\\[2ex] 2 (x-3)x-[x], & x>3\\[2ex] b, & x=3, cases where [x] denotes the greatest integer less than or equal to x. If S denotes the set of all ordered pairs (a, b) such that f(x) is continuous at x=3, then the number of elements in S is :

1. Infinitely many
2. 4
3. 2
4. 1

Let f(x)=x3+x2 f'(1)+x f''(2)+f'''(3), x . Then f'(10) is equal to .