Given the function f (x) = - A & # 178; X & # 179 / / 3 + ax & # 178 / / 2 + CX (a is not equal to 0), when a ≥ 1 / 2, if the real number equation f '(x) = 0 When a ≥ 1 / 2, if the real number equation f '(x) = 0 has two real number roots m, N, and | m | ≤ 1, | n | ≤ 1. Prove: - 1 / 4 ≤ C ≤ A & # - A

Given the function f (x) = - A & # 178; X & # 179 / / 3 + ax & # 178 / / 2 + CX (a is not equal to 0), when a ≥ 1 / 2, if the real number equation f '(x) = 0 When a ≥ 1 / 2, if the real number equation f '(x) = 0 has two real number roots m, N, and | m | ≤ 1, | n | ≤ 1. Prove: - 1 / 4 ≤ C ≤ A & # - A

If the original function is derived, f '(x) = - A & # 178; X & # 178; + ax + C and f' (x) = 0 has two real roots m, N, so △ ≥ 0 means a & # 178; + 4CA & # 178; ≥ 0 → a & # 178; (1 + 4C) ≥ 0 means 1 + 4C ≥ 0, then C ≥ - 1 / 4 is obtained. According to Weida's theorem, Mn = 1 / a (1) m + n = - C / A & # 178