Application of Derivatives

Let f : R R be a differentiable function such that f ( x+y3 )= f(x)+f(y)3 for all x, y R, and f'(0)=3. Then the minimum value of the function g(x)=3+ex f(x), is:

1. 3 ( e+1e )
2. 3 ( e-1e )
3. 3-ee
4. 3e

0 x (16 ( x2 ) 3 ( x2 ) ) is equal to :

1. 3 32
2. 3 3
3. 4 3
4. 6 3

The number of critical points of the function f(x)= cases | xx |, & x 0\ 1, & x=0 cases in the interval (-2 , 2 ) is equal to :

1. 1
2. 3
3. 5
4. 7

Let f(x) be a polynomial of degree 5, and have extrema at x=1 and x=-1. If x 0 ( f(x)x3 )=-5, then f(2)-f(-2) is equal to:

1. 0
2. 50
3. 92
4. 112

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