Application of Derivatives

For the function f(x) = x + 3x - 2 (x2 + x ), where x [0, 2 ], consider the following two statements : (I) f is increasing in (0, 2 ). (II) f' is decreasing in (0, 2 ). Between the above two statements,

1. only (I) is true.
2. both (I) and (II) are true.
3. only (II) is true.
4. neither (I) nor (II) is true.

Let f(x) = 3 x - 2 + 4 - x be a real valued function. If and are respectively the minimum and the maximum values of f, then 2 + 2 2 is equal to

1. 42
2. 38
3. 24
4. 44

Let f : R R be a thrice differentiable function such that f(0) = 0, f(1) = 1, f(2) = -1, f(3) = 2 and f(4) = -2. Then, the minimum number of zeros of (3f'f'' + ff''')(x) is .

If 5f(x)+4f ( 1x )=x2-2, x 0 and y=9x2f(x), then y is strictly increasing in :

1. (0, 1 5 ) ( 1 5, )
2. (- 1 5, 0 ) ( 1 5, )
3. (- 1 5, 0 ) (0, 1 5 )
4. (- , 1 5 ) (0, 1 5 )

Let S=(-1, ) and f : S R be defined as f(x)= -1x (et-1 )11(2t-1)5(t-2)7(t-3)12(2t-10)61dt, Let p= Sum of squares of the values of x, where f(x) attains local maxima on S, and q= Sum of the values of x, where f(x) attains local minima on S. Then, the value of p2+2q is .

If the function f:(- ,-1] (a,b] defined by f(x)=ex3-3x+1 is one-one and onto, then the distance of the point P(2b+4,a+2) from the line x+e-3y=4 is :

1. 2 1+e6
2. 1+e6
3. 3 1+e6
4. 4 1+e6

Let g:R be a non constant twice differentiable function such that g' ( 12 )=g' ( 32 ). If a real valued function f is defined as f(x)= 12[g(x)+g(2-x)], then

1. f''(x)=0 for at least two x in (0,2)
2. f' ( 32 )+f' ( 12 )=1
3. f''(x)=0 for no x in (0,1)
4. f''(x)=0 for exactly one x in (0,1)

The maximum area of a triangle whose one vertex is at (0,0) and the other two vertices lie on the curve y=-2x2+54 at points (x, y) and (-x, y), where y>0, is :

1. 108
2. 122
3. 88
4. 92

Let f(x)=(x+3)2(x-2)3, x [-4,4]. If M and m are the maximum and minimum values of f, respectively in [-4,4], then the value of M-m is

1. 108
2. 392
3. 608
4. 600

In an A.P., the sixth term a6=2. If the product a1a4a5 is the greatest, then the common difference of the A.P. is equal to

1. 23
2. 58
3. 32
4. 85