First Derivative Test

Core Sentence / 核心句

The First Derivative Test classifies a critical number by checking how the sign of the derivative changes across it.

中文理解：临界数只是候选；一阶导在左右怎样变号，才决定局部极大、极小或都不是。

Rule / 规则

$$f^{\prime }:+\to -\Rightarrow \text{local maximum}$$ $$f^{\prime }:-\to +\Rightarrow \text{local minimum}$$

No sign change means neither.

Critical-Number Gate / 临界数

$c$ must be in the domain of $f$, with $f^{\prime }(c)=0$ or $f^{\prime }(c)$ nonexistent.

Watch Out / 易错点

• derivative zero alone does not prove an extremum.
• an undefined point outside the function domain is not a critical number.
• even-multiplicity derivative zeros often do not change sign.

Search and Source Mentions / 搜索与出现位置

Mentioned In / 出现位置

• 微积分BC第6次正课 · First Derivative Test

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Derivatives
• Absolute Extrema
• Second Derivative