Let f(x) = x^3 - 6 x^2 + 3( + 16)x + 5 . If ^* is the maximum integer value of for which the function f(x) is bijective on the set of real numbers R , then the value of f(2) when = ^* is equal to:

Step-by-step solution

For a cubic polynomial to be bijective on R , it must be strictly monotonic, which implies its derivative f'(x) must not change sign. Given f(x) = x^3 - 6 x^2 + 3( + 16)x + 5 Differentiating with respect to x , we get: f'(x) = 3 x^2 - 12 x + 3( + 16) For f(x) to be strictly increasing (since for This means the discriminant D of the quadratic f'(x) must be less than or equal to zero, and 0 . D = (-12 )^2 - 4(3 )(3( + 16)) 0 144 ^2 - 36 ( + 16) 0 144 ^2 - 36 ^2 - 576 0 108 ^2 - 576 0 3 ^2 - 16 0 (3 - 16) 0 Thus, [0,