Limits and Derivatives

Let \x\ denote the fractional part of x and f(x)= -1 (1-\x\2 ) -1(1-\x\)\x\-\x\3, x 0. If L and R respectively denotes the left hand limit and the right hand limit of f(x) at x=0, then 32 2 (L2+R2 ) is equal to .

Let f(x)= casesx-1, & x is even,\ 2x, & x is odd, cases x . If for some a , f(f(f(a)))=21, then x - \ |x|3a- [ xa ] \, where [t] denotes the greatest integer less than or equal to t, is equal to :

1. 169
2. 121
3. 225
4. 144

x 0 e2| x|-2| x|-1x2

1. is equal to 1
2. does not exist
3. is equal to -1
4. is equal to 2

Let f: R (0, ) be a strictly increasing function such that x f(7x)f(x)=1. Then, the value of x [ f(5x)f(x)-1 ] is equal to

1. 0
2. 4
3. 1
4. 7/5

If x 0 ax2ex-b e(1+x)+cxe-xx2 x=1, then 16 (a2+b2+c2 ) is equal to .

Let f: [- 2, 2 ] be a differentiable function such that f(0)= 12. If the x 0 x 0x f(t) dtex2-1= , then 8 2 is equal to :

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

Suppose f(x)= (2x+2-x ) x -1 (x2-x+1 ) (7x2+3x+1 )3. Then the value of f (0) is equal to

3. 0
4. 2

x 2 ( 1 (x- 2 )2 x3 ( 2 )3 (t 13 )dt ) is equal to

1. 3 24
2. 3 28
3. 3 4
4. 3 8

Let the slope of the line 45x+5y+3=0 be 27r1+ 9r22 for some r1, r2 R. Then x 3 ( 3x 8t2 3r2x2-r2x2-r1x3-3x dt ) is equal to .

If a= x 0 1+ 1+x4- 2x4 and b= x 0 2x 2- 1+ x, then the value of ab3 is :

1. 36
2. 25
3. 32
4. 30