Let the range of the function g(x) = 10 - 16 x ( 3 - x ) ( 3 + x ) 3x , where x R , be [a, b] . If a and b are the roots of the quadratic equation x^2 - px + q = 0 , then the value of q - 4p is :

Step-by-step solution

Given function is: g(x) = 10 - 16 x ( 3 - x ) ( 3 + x ) 3x Using the identity x ( 3 - x ) ( 3 + x ) = 1 4 3x , we get: g(x) = 10 - 16 ( 1 4 3x ) 3x g(x) = 10 - 4 3x 3x Using the double angle identity 2 = 2 : g(x) = 10 - 2 6x Since the range of 6x is [-1, 1] , the minimum and maximum values of g(x) are: Minimum value = 10 - 2(1) = 8 Maximum value = 10 - 2(-1) = 12 Thus, the range of g(x) is [8, 12] , which means a = 8 and b = 12 . Since a and b are the roots of x^2 - px + q = 0 : Sum of roots p = a + b = 8 + 12 = 20