如何判断函数极值点与拐点

一、极值点

• 极值的必要条件：f′(a)=0" role="presentation">f′(a)=0$f^{\prime }(a)=0$
• 极值的第一充分条件：f′(a)=0" role="presentation" style="position: relative;">f′(a)=0$f^{\prime }(a)=0$且f′(x)" role="presentation" style="position: relative;">f′(x)$f^{\prime }(x)$在x=a" role="presentation" style="position: relative;">x=a$x=a$两侧变号
• 极值的第二充分条件：f′(a)=0" role="presentation" style="position: relative;">f′(a)=0$f^{\prime }(a)=0$且f″(a)≠0" role="presentation" style="position: relative;">f′′(a)≠0$f^{″}(a)\ne 0$(f″(a)>0" role="presentation" style="position: relative;">f′′(a)>0$f^{″}(a)>0$为极小值，f″(a)<0" role="presentation" style="position: relative;">f′′(a)<0$f^{″}(a)<0$为极大值)
• 极值的第三充分条件：设f(x)" role="presentation" style="position: relative;">f(x)$f(x)$在x=a" role="presentation" style="position: relative;">x=a$x=a$处最低阶不为零的导数的阶为n" role="presentation" style="position: relative;">n$n$，若n" role="presentation" style="position: relative;">n$n$为偶数x=a" role="presentation" style="position: relative;">x=a$x=a$是极值点。若n" role="presentation" style="position: relative;">n$n$为奇数x=a" role="presentation" style="position: relative;">x=a$x=a$是不是极值点

二、拐点

函数的拐点可理解为导数的极值点，因此上述关于极值点的结论都可“稍加改变”后用于判断拐点，下面是一些常用结论：

• 拐点的必要条件：f″(a)=0" role="presentation" style="position: relative;">f′′(a)=0$f^{″}(a)=0$
• 拐点的充分条件：f″(a)=0" role="presentation" style="position: relative;">f′′(a)=0$f^{″}(a)=0$且f′(x)" role="presentation" style="position: relative;">f′(x)$f^{\prime }(x)$在x=a" role="presentation" style="position: relative;">x=a$x=a$左右两侧变号
• 利用三阶导数的判别法：f′(a)=f″(a)=0" role="presentation" style="position: relative;">f′(a)=f′′(a)=0$f^{\prime }(a)=f^{″}(a)=0$，f‴(a)≠0" role="presentation" style="position: relative;">f′′′(a)≠0$f^{‴}(a)\ne 0$

三、情形分析

情形一：，

• x=a" role="presentation" style="position: relative;">x=a$x=a$既不是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点也不是拐点。例如一次函数f(x)=2x" role="presentation" style="position: relative;">f(x)=2x$f(x)=2x$，有f′(0)=2" role="presentation" style="position: relative;">f′(0)=2$f^{\prime }(0)=2$，f″(0)=0" role="presentation" style="position: relative;">f′′(0)=0$f^{″}(0)=0$，但显然x=0" role="presentation" style="position: relative;">x=0$x=0$既不是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点也不是拐点
• x=a" role="presentation" style="position: relative;">x=a$x=a$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的拐点，例如f(x)=x3+x" role="presentation" style="position: relative;">f(x)=x3+x$f(x)=x^3+x$，由于f′(0)=1" role="presentation" style="position: relative;">f′(0)=1$f^{\prime }(0)=1$，f″(0)=0" role="presentation" style="position: relative;">f′′(0)=0$f^{″}(0)=0$，f‴(0)=6" role="presentation" style="position: relative;">f′′′(0)=6$f^{‴}(0)=6$，故x=0" role="presentation" style="position: relative;">x=0$x=0$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的拐点

情形二：，

• x=a" role="presentation" style="position: relative;">x=a$x=a$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点，例如f(x)=x2" role="presentation" style="position: relative;">f(x)=x2$f(x)=x^2$，满足f′(0)=0" role="presentation" style="position: relative;">f′(0)=0$f^{\prime }(0)=0$，f″(0)=2" role="presentation" style="position: relative;">f′′(0)=2$f^{″}(0)=2$，显然x=0" role="presentation" style="position: relative;">x=0$x=0$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极(小)值点

情形三：，

• x=a" role="presentation" style="position: relative;">x=a$x=a$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点。例如f(x)=x4" role="presentation" style="position: relative;">f(x)=x4$f(x)=x^4$满足f′(0)=f″(0)=0" role="presentation" style="position: relative;">f′(0)=f′′(0)=0$f^{\prime }(0)=f^{″}(0)=0$，显然x=0" role="presentation" style="position: relative;">x=0$x=0$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极小值点
• x=a" role="presentation" style="position: relative;">x=a$x=a$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的拐点。例如f(x)=x3" role="presentation" style="position: relative;">f(x)=x3$f(x)=x^3$，满足f′(0)=f″(0)=0" role="presentation" style="position: relative;">f′(0)=f′′(0)=0$f^{\prime }(0)=f^{″}(0)=0$，显然x=0" role="presentation" style="position: relative;">x=0$x=0$是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的拐点
• x=a" role="presentation" style="position: relative;">x=a$x=a$既不是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点也不是拐点。例如f(x)=C" role="presentation" style="position: relative;">f(x)=C$f(x)=C$(常值函数)，显然任意点处一、二阶导数都等于0，但f(x)" role="presentation" style="position: relative;">f(x)$f(x)$既无极值点也无拐点

情形四：，

• 这是平凡的情形，显然x=a" role="presentation" style="position: relative;">x=a$x=a$既不是f(x)" role="presentation" style="position: relative;">f(x)$f(x)$的极值点也不是拐点。