Question

已知f(x)等于x^3+ax^2+bx+c,当x等于-1时，f（x)的极大值为7，x等于3时，f(x）有极小值。a，b，c的值。函数

Answer 1

f`(x)=3x^2+2ax+b
-1+3=2=-2a/3
a=-3
-1*3=b/3
b=-9
f(x)=x^3-3x^2-9x+c
x=-1
7=-1-3+9+c
c=2

Answer 2

f(x)= x^3+ax^2+bx+c
f'(x) = 3x^2+ 2ax+b
f'(-1) =0
=> 3 - 2a + b = 0
b = 2a-3 (1)
f'(3) =0
27 +6a + b = 0
b = -27-6a (2)

from (1) and (2)
2a-3 = -27-6a
8a = -24
a = -3
b = 2(-3) -3 = -9

f(x) = x^3-3 x^2-9x+ c
f(-1) = -1 -3 + 9 + c = 7
=> c = 2
f(x) = x^3-3x^2-9x+ 2
f(3) = 27 - 27 - 27 + 7
= -20

Answer 3

解：由题意得
f'(x)=3x^2+2ax+b
则有f(-1)=7 a=-3
f'(-1)=0 解得 b=-9
f'(3)= 0 c=-5