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9.已知f(x)=ax5-bx3+c(a>0)在x=±1處有極值,且極大值為4,極小值為0,試確定a、b、c的值.
解:已知f(x)=ax5-bx3+c,
所以(x)=5ax4-3bx2=x2(5ax2-3b).
根據(jù)題意(x)=0應有根x=±1,
故5a=3b.
所以(x)=5ax2(x2-1).
因a>0時,列表:
x |
(-∞,-1) |
-1 |
(-1,1) |
1 |
(1,+∞) |
(x) |
+ |
0 |
- |
0 |
+ |
f(x) |
|
極大值 |
|
極小值 |
|
|
①+②得c=2,
①-②得b=a+2.
又5a=3b,所以a=3,b=5,c=2.
探究創(chuàng)新