Concavity

Core Sentence / 核心句

Concavity describes whether tangent slopes increase or decrease as the input moves to the right.

中文理解：凹凸性不是看函数上升还是下降，而是看切线斜率从左到右增加还是减少。

Formula / 公式或规则

$$f^{\prime \prime }(x)>0\Rightarrow \text{concave up}$$ $$f^{\prime \prime }(x)<0\Rightarrow \text{concave down}$$

Inflection Point / 拐点

An inflection point is a point where concavity changes.

$f^{\prime \prime }(c)=0$ 只是 candidate condition；必须检查 $c$ 两侧 concavity 是否改变。

Watch Out / 易错点

increasing/decreasing 由 $f^{\prime }$ 判断；concavity 由 $f^{\prime \prime }$ 判断。两者不要混用。

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 · Concavity
  • the sign of f” tells whether tangent slopes increase or decrease
• Chapter 10 · Concavity and Sign
• Chapter 10 · Magnitude and Inflection Warning

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Inflection Points
• 10 - Higher Order Derivatives
• Second Derivative
• Derivatives
• Taylor Series