Application of Derivatives

Let the set of all values of p, for which f(x) = (p2 - 6p + 8 ) ( 2 2x - 2 2x ) + 2(2 - p)x + 7 does not have any critical point, be the interval (a, b). Then 16ab is equal to .

Let f(x) = 4 3 x + 3 3 2 x - 10. The number of points of local maxima of f in interval (0, 2 ) is

1. 1
2. 3
3. 4
4. 2

The number of critical points of the function f(x) = (x - 2)2/3(2x + 1) is

1. 2
2. 1
3. 0
4. 3

For the function f(x) = ( x) - x + 1, x R, between the following two statements (S1) f(x) = 0 for only one value of x in [0, ]. (S2) f(x) is decreasing in [0, 2 ] and increasing in [ 2, ].

1. Both (S1) and (S2) are incorrect.
2. Only (S1) is correct.
3. Only (S2) is correct.
4. Both (S1) and (S2) are correct.

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.