Let a and b respectively be the points of local maximum and local minimum of the function f ( x ) = 2 x 3 - 3 x 2 - 12 x . If A is the total area of the region bounded by y = f ( x ) , the x -axis and the lines x = a and x = b , then 4 A is equal to ______.

Step-by-step solution

. f ' ( x ) = 6 x 2 - 6 x - 12 = 6 x 2 - x - 2 = 6 ( x - 2 ) ( x + 1 ) Clearly, f ' - 1 - > 0   &   f ' - 1 + < 0 Hence, x = - 1 is the point of local maximum. Also, f ' 2 - < 0   &   f ' 2 + > 0 Hence, x = 2 is the point of local minimum. ∴ a = - 1   &   b = 2 Points= 2 , - 20   &   - 1 , 7 A = ∫ - 1 0 2 x 3 - 3 x 2 - 12 x d x - ∫ 0 2 2 x 3 - 3 x 2 - 12 x d x = x 4 2 - x 3 - 6 x 2 - 1 0 - x 4 2 - x 3 - 6 x