Application of Derivatives

If the function f(x) = 2x3 - 9ax2 + 12a2x + 1, a > 0 has a local maximum at x = and a local minimum at x = 2, then and 2 are the roots of the equation :

1. x2 - 6x + 8 = 0
2. 8x2 - 6x + 1 = 0
3. 8x2 + 6x - 1 = 0
4. x2 + 6x + 8 = 0

Let A be the region enclosed by the parabola y2 = 2x and the line x = 24. Then the maximum area of the rectangle inscribed in the region A is .

The interval in which the function f(x) = xx, x > 0, is strictly increasing is

1. (0, )
2. (0, 1e ]
3. [ 1e2, 1 )
4. [ 1e, )

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 .