Extreme Value Theorem

Core Sentence / 核心句

A continuous function on a closed bounded interval must attain an absolute maximum and an absolute minimum.

中文理解：闭有界区间上的连续函数一定真正取到最高值和最低值。

Hypothesis / 前提

$$f\text{ continuous on }[a,b].$$

Conclusion / 结论

There exist $x_{\min },x_{\max }\in [a,b]$ such that

$$f(x_{\min })\le f(x)\le f(x_{\max })$$

for all $x\in [a,b]$。

Watch Out / 易错点

• EVT guarantees existence, not location.
• closed, bounded, and continuous all matter.
• EVT is not the calculation method; use Absolute Extrema to find values.

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 · EVT

Connected Notes / 连接

• 微积分BC第6次正课-导数符号极值与凹凸性
• Absolute Extrema
• Intermediate Value Theorem
• Mean Value Theorem