Let S be the set of all integer values of the parameter a for which the function f(x) = 2x^3 - 3(a^2 + a + 3)x^2 + 18(a^2 + a)x - 25 is strictly decreasing on the interval (1, 3) . The sum of the squares of the elements in S is:

Step-by-step solution

Given f(x) = 2x^3 - 3(a^2 + a + 3)x^2 + 18(a^2 + a)x - 25 Differentiating with respect to x : f'(x) = 6x^2 - 6(a^2 + a + 3)x + 18(a^2 + a) We can factor the quadratic expression: f'(x) = 6[x^2 - (a^2 + a + 3)x + 3(a^2 + a)] f'(x) = 6(x - (a^2 + a))(x - 3) For f(x) to be strictly decreasing on the interval (1, 3) , we must have f'(x) 0 for all x (1, 3) . The roots of f'(x) = 0 are x = a^2 + a and x = 3 . Since the leading coefficient of f'(x) is positive, f'(x) 0 between its roots. Thus, the interval where f(x) is d