Application of Derivatives

If the function f(x) = ( 1x )2x; x > 0 attains the maximum value at x = 1e then :

1. e < e
2. e2 < (2 )e
3. (2e) > (2e)
4. e > e

Let a rectangle ABCD of sides 2 and 4 be inscribed in another rectangle PQRS such that the vertices of the rectangle ABCD lie on the sides of the rectangle PQRS. Let a and b be the sides of the rectangle PQRS when its area is maximum. Then (a + b)2 is equal to :

1. 64
2. 80
3. 60
4. 72

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