Extreme Value Theorem, Global Versus Local Extrema, and Critical Points

Peaks, valleys, and where to look for them

• A smooth graph has high points and low points — the derivative is the tool for finding them.
• First, some vocabulary: a local (relative) extremum 局部极值 is a peak or valley compared to nearby points.
• A global (absolute) extremum 全局极值 is the single highest or lowest over the whole interval.
• The candidates for both are special inputs called critical points 临界点.

The Extreme Value Theorem

• The Extreme Value Theorem 极值定理 (EVT) guarantees extrema exist — under a condition.
• If $f$ is continuous on a closed interval $[a,b]$, then $f$ attains an absolute max and an absolute min on $[a,b]$.
• Continuity + closed interval is the entry ticket; drop either and a max or min may not exist.
• The EVT tells you they exist; the derivative helps you find them.

At a differentiable interior extremum the derivative is zero.

Critical points: where extrema can hide

• A critical point is an interior $x$ where $f'(x)=0$ or $f'(x)$ does not exist.
• $f'=0$: a horizontal tangent (a smooth peak or valley). $f'$ undefined: a corner or cusp.
• Local extrema can occur only at critical points — nowhere else inside the interval.
• So to hunt extrema, first solve $f'(x)=0$ and note where $f'$ is undefined.

Local vs. global — don't confuse them

• A local max is taller than its immediate neighbors; a global max is the tallest anywhere on $[a,b]$.
• A global extremum on a closed interval sits at either a critical point or an endpoint.
• Not every critical point is an extremum (a horizontal tangent can be a flat spot on a rising curve).
• Local and global are different questions — a point can be one without being the other.

Write from a candidate that is not a winner

• A function is continuous on a closed interval.
• f prime is 0 at an interior point, but that point is a flat shelf, not a peak.