Application of Derivatives

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 .

The function f(x)= xx2-6x-16, x R-\-2,8\

1. decreases in (- ,-2) (-2,8) (8, )
2. increases in (- ,-2) (-2,8) (8, )
3. decreases in (-2,8) and increases in (- ,-2) (8, )
4. decreases in (- ,-2) and increases in (8, )

The function f(x)=2x+3(x) 23, x R, has

1. exactly one point of local minima and no point of local maxima
2. exactly one point of local maxima and exactly one point of local minima
3. exactly two points of local maxima and exactly one point of local minima
4. exactly one point of local maxima and no point of local minima

Let g(x)=3f ( x3 )+f(3-x) and f''(x)>0 for all x (0,3). If g is decreasing in (0, ) and increasing in ( ,3), then 8 is :

1. 0
2. 24
3. 18
4. 20