Application of Derivatives

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

Let f(x)= 0x t (t2-9t+20 )dt, 1 x 5. If the range of f is [ , ], then 4( + ) equals :

1. 253
2. 157
3. 154
4. 125

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

Consider the region R= \(x,y):x y 9- 113x2, x 0 \. The area of the largest rectangle of sides parallel to the coordinate axes and inscribed in R, is:

1. 821123
2. 567121
3. 730119
4. 625111

Let (2,3) be the largest open interval in which the function f(x)=2 e(x-2)-x2+ax+1 is strictly increasing and (b,c) be the largest open interval, in which the function g(x)=(x-1)3(x+2-a)2 is strictly decreasing. Then 100(a+b-c) is equal to :

1. 360
2. 420
3. 160
4. 280

If the set of all values of a, for which the equation 5x3-15x-a=0 has three distinct real roots, is the interval ( , ), then -2 is equal to .

A spherical chocolate ball has a layer of ice-cream of uniform thickness around it. When the thickness of the ice-cream layer is 1 cm, the ice-cream melts at the rate of 81 cm3/min and the thickness of the ice-cream layer decreases at the rate of 14 cm/min. The surface area (in cm2) of the chocolate ball (without the ice-cream layer) is :

1. 128
2. 196
3. 225
4. 256