Given the function f (x) = ax & # 178; + 1 (a > 0), G (x) = x ^ 3 + BX (1) If the curve y = f (x) and the curve y = g (x) have a common tangent at their intersection (1, c), the values of a and B are obtained; (2) When a ^ 2 = 4b, find the monotone interval of F (x) + G (x) - and its maximum value in the interval (- ∞, - 1)

Given the function f (x) = ax & # 178; + 1 (a > 0), G (x) = x ^ 3 + BX (1) If the curve y = f (x) and the curve y = g (x) have a common tangent at their intersection (1, c), the values of a and B are obtained; (2) When a ^ 2 = 4b, find the monotone interval of F (x) + G (x) - and its maximum value in the interval (- ∞, - 1)

1. First of all, we obtain the derivatives of F (x) g (x), and then substitute x = 1 to get an equation: 2A = 3 + B (1); then we substitute (1, c) into two lions to get two equations: a + 1 = C (2); 1 + B = C (3); simultaneous (2) (3) can get a = B; then a = b = 3; then a = b = 3;
2. First of all, substitute a ^ 2 = 4B into G (x), then G (x) + F (x) = x ^ 3 + ax ^ 2 + (a ^ 2 / 4) x + 1; let the above function be f (x), then take the derivative of F (x), and get 3x ^ 2 + 2aX + (a ^ 2 / 4) = 0; the solution is X1 = - (A / 2); x2 = - (A / 6); that is, decrease (- A / 2, - A / 6) and increase; when 6 > a > 2, f (- A / 2) is the maximum, you can list the rest by yourself. I'm too lazy to play