Continuity and Differentiability

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 .

Let for a differentiable function f:(0, ) , f(x)-f(y) e ( xy )+x-y, x, y (0, ). Then n=120f' ( 1n2 ) is equal to .

Consider the function f:(0,2) defined by f(x)= x2+ 2x and the function g(x) defined by g(x)= cases \f(t)\, & 0<t x and 0<x 1\ 32+x, & 1<x<2 cases. Then,

1. g is continuous but not differentiable at x=1
2. g is continuous and differentiable for all x (0,2)
3. g is not continuous for all x (0,2)
4. g is neither continuous nor differentiable at x=1