4.4 微分中值定理

设 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 的某一个邻域 ${\displaystyle o(x_0)}$ 内有定义, 且 ${\displaystyle \mathrm{\forall }x\in o(x_0):f(x)\le f(x_0)}$, 则称 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 上取到极大值, 称 ${\displaystyle x_0}$ 是 ${\displaystyle f(x)}$ 的极大值点. 若不等式改为 ${\displaystyle f(x)\ge f(x_0)}$, 则称 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 上取到极小值, 称 ${\displaystyle x_0}$ 是 ${\displaystyle f(x)}$ 的极小值点.
若将 ${\displaystyle f(x)\le f(x_0)}$ 改为 ${\displaystyle f(x)<f(x_0)}$, 则称 ${\displaystyle x_0}$ 是 ${\displaystyle f(x)}$ 的严格极大值点, 同理可以定义严格极小值点.

称 ${\displaystyle f^{\prime }(x)}$ 的零点为 ${\displaystyle f(x)}$ 的驻点.

若 ${\displaystyle x_0}$ 是 ${\displaystyle f(x)}$ 的极值点, 且 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 处可导, 则 ${\displaystyle f^{\prime }(x_0)=0}$.

若 ${\displaystyle f(x)\in C[a,b],f(x)\in D(a,b)}$, 且 ${\displaystyle f(a)=f(b)}$, 则 ${\displaystyle \mathrm{\exists }\xi \in (a,b):f^{\prime }(\xi )=0}$.

若 ${\displaystyle f(x)\in C[a,b],f(x)\in D(a,b)}$, 则 ${\displaystyle \mathrm{\exists }\xi \in (a,b)}$:

若 ${\displaystyle f(t),g(t)\in C[a,b],f(t),g(t)\in D(a,b)}$, 且 ${\displaystyle \mathrm{\forall }t\in (a,b):g^{\prime }(t)\ne 0}$, 则 ${\displaystyle \mathrm{\exists }\xi \in (a,b)}$:

设 ${\displaystyle f(x)\in D[a,b]}$, ${\displaystyle f_{+}^{\prime }(a)<f_{-}^{\prime }(b)}$, 则 ${\displaystyle \mathrm{\forall }\lambda \in (f_{+}^{\prime }(a),f_{-}^{\prime }(b))}$, ${\displaystyle \mathrm{\exists }\xi \in (a,b):f^{\prime }(\xi )=\lambda }$.

若 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 处连续, 在 ${\displaystyle (x_0,x_0+\delta )}$ 内可导, 且导函数存在右极限 ${\displaystyle f^{\prime }(x_0^{+})}$, 则 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 处右可导, 且 ${\displaystyle f_{+}^{\prime }(x_0)=f^{\prime }(x_0^{+})}$.

若 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 处连续, 在 ${\displaystyle o({\hat{x}}_0,\delta )}$ 内可导, 且 ${\displaystyle \underset{x\to x_0}{lim}{f}^{\prime }(x)}$ 存在, 则 ${\displaystyle f(x)}$ 在 ${\displaystyle x_0}$ 处可导, 且 ${\displaystyle f^{\prime }(x_0)=\underset{x\to x_0}{lim}{f}^{\prime }(x)}$.

设 ${\displaystyle f(x)\in D(a,b)}$, 则 ${\displaystyle \mathrm{\forall }{x}_0\in (a,b)}$, ${\displaystyle x_0}$ 或者是 ${\displaystyle f^{\prime }(x)}$ 的连续点, 或者是第二类间断点.