Application of Derivatives

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

Consider the function f: [ 12,1 ] R defined by f(x)=4 2x3-3 2x-1. Consider the statements (I) The curve y=f(x) intersects the x-axis exactly at one point. (II) The curve y=f(x) intersects the x-axis at x= 12. Then

1. Both (I) and (II) are correct.
2. Only (I) is correct.
3. Both (I) and (II) are incorrect.
4. Only (II) is correct.

Let f(x)=2x-x2,x R. If m and n are respectively the number of points at which the curves y=f(x) and y=f'(x) intersect the x-axis, then the value of m+n is .