Question

9．函數(shù)f（x）=x³-ax²-bx+a²在x=1處有極值10，則a，b的值為（　�。�

• A． $\left\{\begin{array}{l}{a=3}\\{b=-3}\end{array}\right.$或$\left\{\begin{array}{l}{a=-4}\\{b=11}\end{array}\right.$
• B． $\left\{\begin{array}{l}{a=-4}\\{b=11}\end{array}\right.$
• C． $\left\{\begin{array}{l}{a=-1}\\{b=5}\end{array}\right.$
• D． 以上都不對(duì)

Answer 1

分析 利用函數(shù)值，函數(shù)的導(dǎo)數(shù)列出方程求解即可．

解答 解：函數(shù)f（x）=x³-ax²-bx+a²，f′（x）=3x²-2ax-b，
函數(shù)f（x）=x³-ax²-bx+a²在x=1處有極值10，
可得：$\left\{\begin{array}{l}3-2a-b=0\\ 1-a-b+{a}^{2}=10\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=3}\\{b=-3}\end{array}\right.$或$\left\{\begin{array}{l}{a=-4}\\{b=11}\end{array}\right.$，
當(dāng)$\left\{\begin{array}{l}{a=3}\\{b=-3}\end{array}\right.$時(shí)，f′（x）=3x²-6x+3≥0恒成立，x=1不是極值點(diǎn)．
當(dāng)$\left\{\begin{array}{l}{a=-4}\\{b=11}\end{array}\right.$時(shí)，f′（x）=3x²+8x-11，△=196＞0，導(dǎo)函數(shù)有兩個(gè)解，x=1是極值點(diǎn)．滿足題意；
故選：B．

點(diǎn)評(píng) 本題考查函數(shù)的導(dǎo)數(shù)的應(yīng)用，函數(shù)的極值的應(yīng)用，注意驗(yàn)證，否則容易錯(cuò)選A．