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

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 : 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

Let f : R R be a function defined by f(x) = ||x + 2| - 2|x| |. If m is the number of points of local minima and n is the number of points of local maxima of f, then m + n is

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

If the function f(x) = 2x3 - 9ax2 + 12a2x + 1, where a > 0, attains its local maximum and local minimum values at p and q, respectively, such that p2 = q, then f(3) is equal to :

1. 55
2. 37
3. 10
4. 23

The sum of all local minimum values of the function f(x) = cases 1 - 2x, & x < -1 \ 13(7 + 2|x|), & -1 x 2 \ 1118(x - 4)(x - 5), & x > 2 cases is

1. 16772
2. 15772
3. 17172
4. 13172

Let f(x)= 0x2 t2-8t+15et dt, x . Then the numbers of local maximum and local minimum points of f, respectively, are :

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

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

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

In an A.P., the sixth term a6=2. If the product a1a4a5 is the greatest, then the common difference of the A.P. is equal to

1. 23
2. 58
3. 32
4. 85

The function f(x)=2x+3(x) 23, x R, has

1. exactly one point of local minima and no point of local maxima
2. exactly one point of local maxima and exactly one point of local minima
3. exactly two points of local maxima and exactly one point of local minima
4. exactly one point of local maxima and no point of local minima