Limits and Derivatives

Let and be the real numbers such that x 0 1x3 ( 2 0x 11 - t2 dt + x x ) = 2. Then the value of + is .

Let x0 be the real number such that ex0+x0=0. For a given real number , define g(x)= 3xex+3x- ex- x3(ex+1) for all real numbers x. Then which one of the following statements is TRUE?

1. For =2, x x0 | g(x)+ex0x-x0 |=0
2. For =2, x x0 | g(x)+ex0x-x0 |=1
3. For =3, x x0 | g(x)+ex0x-x0 |=0
4. For =3, x x0 | g(x)+ex0x-x0 |= 23

Let k R. If x 0+ ( ( kx)+ x+x ) 2x=e6, then the value of k is

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

Let S be the set of all ( , ) R R such that x (x2)( e x) ( 1x2 )x ( e(1+x)) =0. Then which of the following is (are) correct?

1. (-1,3) S
2. (-1,1) S
3. (1,-1) S
4. (1,-2) S