一、题目![题目 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/f68a9e590526998388b0f9b71bd5d3f73dda4ed9764819fe8f36488fa537e9b9499f465fd201d7c117b8901c3ad071915a34a688058a739ebc39835753a8d7cc.svg)
已知,$(1,3)$ 是曲线 $y=x^{3}+a x^{2}+b x+14$ 的拐点,则 $a=?$, $b=?$
难度评级:
二、解析 ![解析 - 荒原之梦](https://documents.zhaokaifeng.com/uploads/2017/06/06/6fff698aa5c66c6c7a143e3d2a00fa8ee7eab76be5360d89eb43a03143848e8cd60377c76bf830c93ec6603be5af661d9c52238834792ea548bf14de10b05ad9.svg)
错误的解法(误认为拐点是一阶导等于零的点)
$$
y=x^{3}+a x^{2}+b x+14 \Rightarrow
$$
$$
y^{\prime}=3 x^{2}+2 a x+b \Rightarrow
$$
$$
\left\{\begin{array}{l}y^{\prime}(1)=0 \Rightarrow 3+2 a+b=0 \\ y(1)=3 \Rightarrow 1+a+b+14=3 \end{array}\right.
\Rightarrow
$$
$$
\left\{\begin{array}{l}3+2 a+b=0 \\ a+b=3-15 \end{array}\right. \quad \Rightarrow
$$
$$
\left\{\begin{array}{l}3+2 a+b=0 \\ a=-b-12\end{array}\right. \Rightarrow
$$
$$
3-2(b+12)+b=0 \Rightarrow
$$
$$
3-2 b-24+b=0 \Rightarrow b=-21 \Rightarrow
$$
$$
a=9 \Rightarrow
$$
$$
\left\{\begin{array}{l}a=9 \\ b=-21\end{array}\right.
$$
正确的解法
$$
y=x^{3}+a x^{2}+b x+14 \Rightarrow
$$
$$
y^{\prime}=3 x^{2}+2 a x+b \Rightarrow
$$
$$
y^{\prime \prime}=6 x+2 a \Rightarrow
$$
$$
y^{\prime \prime}(1)=0 \Rightarrow 6+2 a=0 \Rightarrow \textcolor{springgreen}{ a=-3 }
$$
$$
y=x^{3}-3 x^{2}+b x+14 \Rightarrow
$$
$$
y(1)=3 \Rightarrow \textcolor{orangered}{ 1-3+b+14=3 } \Rightarrow
$$
$$
b+15 – 3=3 \Rightarrow
$$
$$
b=3-12 \Rightarrow \textcolor{springgreen}{ b = -9 }
$$
注意:
$1 \textcolor{orangered}{ -3 } +b+14$ $=$ $\textcolor{orangered}{ 3 }$ $\nRightarrow$ $1 + 0+b+14=0$