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问题 解答题
函数f(x)=
1
3
x3-ax2+(a2-1)x+b(a，b∈R)
（1）若x=1为f（x）的极值点，求a的值．
（2）若y=f（x）的图象在（1，f（1））处的切线方程为x+y-3=0，求f（x）在区间[-2，4]上的最大值．
答案

（1）求导函数可得f′（x）=x2-2ax+a2-1

∵x=1是f（x）的极值点，∴f′（1）=0，∴a2-2a=0，∴a=0或2

（2）∵（1，f（1））在x+y-3=0 上，∴f（1）=2

∵（1，2）在y=f（x）的图象上，∴2=

1
3
-a+a2-1+b

又∵f′（1）=-1，∴1-2a+a2-1=-1，∴a2-2a+1=0

∴a=1，b=

8
3

∴f(x)=

1
3
x3-x2+
8
3

∴f′（x）=x2-2x

∴由f′（x）=0，可知x=0和x=2 是f（x） 的极值点

∵f(0)=

8
3
，f(2)=
4
3
，f（-2）=-4，f（4）=8

∴f（x）在区间[-2，4]上的最大值为8

单项选择题

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