Limits and Derivatives · Question 6

If the function f(x) = ex (e x - x - 1 ) + e( x + x) - x x - x is continuous at x = 0, then the value of f(0) is equal to

1. 23
2. 12
3. 2
4. 32

If x 0 e(a - 1)x + 2 bx + (c - 2)e-xx x - e(1 + x) = 2, then a2 + b2 + c2 is equal to :

1. 3
2. 5
3. 9
4. 7

Let f : R (0, ) be a twice differentiable function such that f(3) = 18, f'(3) = 0 and f''(3) = 4. Then x 1 e ( f(2 + x)f(3) ) 18(x - 1)2 is equal to :

1. 9
2. 18
3. 1
4. 2

Let f(x) = x3 + x2 f'(1) + 2x f''(2) + f'''(3), x R. Then the value of f'(5) is :

1. 6575
2. 1175
3. 25
4. 625

Given below are two statements: In the light of the above statements, choose the correct answer from the options given below:

1. Statement I is false but Statement II is true
2. Both Statement I and Statement II are false
3. Both Statement I and Statement II are true
4. Statement I is true but Statement II is false

x 0+ (5(x) 13 ) e (1 + 3x2 ) ( -13 x )2 (e5(x) 43 - 1 ) is equal to

1. 53
2. 1
3. 13
4. 115

If x 1+ (x - 1)(6 + (x - 1)) + (1 - x)(x - 1)3 = -1, where , R, then + is equal to

1. 20
2. 19
3. 18
4. 17

Let f be a differentiable function on R such that f(2)=1, f'(2)=4. Let x 0(f(2+x))3/x=e . Then the number of times the curve y=4x3-4x2-4( -7)x- meets x-axis is :

1. 3
2. 1
3. 2
4. 0

If x 0 ( xx ) 1x2 = p, then 96 e p is equal to

For , , R, if x 0 x2 x + ( - 1)ex2 2x - x = 3, then + - is equal to :

1. -1
2. 4
3. 6
4. 7