Question

15．若函數(shù)f（x）=x³+ax²+bx+a²在x=1時(shí)有極值10，則實(shí)數(shù)a，b的值是（　�。�

• A． $\left\{{\begin{array}{l}{a=-3}\\{b=3}\end{array}}\right.$
• B． $\left\{{\begin{array}{l}{a=4}\\{b=-11}\end{array}}\right.$
• C． $\left\{{\begin{array}{l}{a=-3}\\{b=3}\end{array}}\right.$或$\left\{{\begin{array}{l}{a=4}\\{b=-11}\end{array}}\right.$
• D． $\left\{{\begin{array}{l}{a=-3}\\{b=-11}\end{array}}\right.$或$\left\{{\begin{array}{l}{a=4}\\{b=3}\end{array}}\right.$

Answer 1

分析 函數(shù)f（x）=x³+ax²+bx+a²在x=1時(shí)有極值10，可得$\left\{\begin{array}{l}{f（1）=1+a+b+{a}^{2}=10}\\{{f}^{′}（1）=3+2a+b=0}\end{array}\right.$，解出即可．

解答 解：f′（x）=3x²+2ax+b，
∵函數(shù)f（x）=x³+ax²+bx+a²在x=1時(shí)有極值10，
∴$\left\{\begin{array}{l}{f（1）=1+a+b+{a}^{2}=10}\\{{f}^{′}（1）=3+2a+b=0}\end{array}\right.$，
解得$\left\{\begin{array}{l}{a=4}\\{b=-11}\end{array}\right.$，
經(jīng)過驗(yàn)證滿足題意．
故選：B．

點(diǎn)評 本題考查了利用導(dǎo)數(shù)研究函數(shù)的單調(diào)性極值，考查了推理能力與計(jì)算能力，屬于中檔題．