Application of Derivatives · Question 6

Let f be a differentiable function satisfying f(x) = 1 - 2x + 0x e(x-t) f(t) dt, x R and let g(x) = 0x (f(t) + 2)15(t - 4)6(t + 12)17 dt, x R. If p and q are respectively the points of local minima and local maxima of g, then the value of |p + q| is equal to .

Let (2 , ) be the largest interval in which the function f(t) = |t+1|t2, t < 0, is strictly decreasing. Then the local maximum value of the function g(x) = 2 e(x-2) + x2 + 4x - , x > 2, is

Consider the following three statements for the function f : (0, ) R defined by f(x) = | e x| - |x - 1| : (I) f is differentiable at all x > 0. (II) f is increasing in (0, 1). (III) f is decreasing in (1, ). Then

1. Only (I) is TRUE.
2. Only (I) and (III) are TRUE.
3. Only (II) and (III) are TRUE.
4. All (I), (II) and (III) are TRUE.

Let f(x) = x2025 - x2000, x [0, 1] and the minimum value of the function f(x) in the interval [0, 1] be (80)80(n)-81. Then n is equal to

1. -40
2. -41
3. -80
4. -81

Let f : R R be a twice differentiable function such that the quadratic equation f(x)m2 - 2f'(x)m + f''(x) = 0 in m, has two equal roots for every x R. If f(0) = 1, f'(0) = 2, and ( , ) is the largest interval in which the function f( e x - x) is increasing, then + is equal to .

Let f : R R be a twice differentiable function such that f''(x) > 0 for all x R and f'(a - 1) = 0, where a is a real number. Let g(x) = f( 2 x - 2 x + a), 0 < x < 2. Consider the following two statements: (I) g is increasing in (0, 4 ) (II) g is decreasing in ( 4, 2 ) Then,

1. Both (I) and (II) are True
2. Neither (I) nor (II) is True
3. Only (I) is True
4. Only (II) is True

Let the function f(x) = x3 + 3x + 3, x 0 be strictly increasing in (- , 1) ( 2, ) and strictly decreasing in ( 3, 4) ( 4, 5). Then i=15 i2 is equal to

1. 48
2. 40
3. 36
4. 28

Let x = -1 and x = 2 be the critical points of the function f(x) = x3 + ax2 + b e |x| + 1, x 0. Let m and M respectively be the absolute minimum and the absolute maximum values of f in the interval [-2, - 12 ]. Then |M + m| is equal to (Take e 2 = 0.7):

1. 21.1
2. 19.8
3. 22.1
4. 20.9

Let f : R R be a polynomial function of degree four having extreme values at x = 4 and x = 5. If x 0 f(x)x2 = 5, then f(2) is equal to :

1. 8
2. 10
3. 12
4. 14

Let a > 0. If the function f(x) = 6x3 - 45ax2 + 108a2x + 1 attains its local maximum and minimum values at the points x1 and x2 respectively such that x1 x2 = 54, then a + x1 + x2 is equal to :

1. 15
2. 13
3. 24
4. 18