Question

求a,b,c的值 ,使f(x)=ax^3+bx^2+cx有一拐点(1,2)且在该点的切线斜率为-1

切线斜率的条件怎么用

Answer 1

f(x) = ax³ + bx² + cx, f(1) = a + b + c = 2 (1)
f'(x) = 3ax² + 2bx + c, f'(1) = 3a + 2b + c = -1 (2) (切线斜率)
f"(x) = 6ax + 2b, f"(1) = 6a + 2b = 0 (拐点) (3)
由(1)-(3): a = 3, b = -9, c = 8