Extreme Value Theorem, Global Versus Local Extrema, and Critical Points
Introduced| English |
|---|
| local (relative) extremum/ˈləʊkl ekˈstreməm/ |
| global (absolute) extremum/ˈɡləʊbl ekˈstreməm/ |
| critical points/ˈkrɪtɪkl pɔɪnts/ |
| Extreme Value Theorem/ekˈstriːm ˈvæljuː ˈθɪərəm/ |
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.
A "local (relative) maximum" is a point that is highest compared to...
Local = compared to nearby points; global = over the whole interval.
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.
The EVT guarantees a max and min when $f$ is continuous on a ____ interval.
Continuity plus a closed interval is required.
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.
Where the slope is zero
y = ax³ + cx
At a local peak or valley the tangent is horizontal ($f'=0$) — those inputs are the critical points.
A critical point is an interior $x$ where...
Critical points: $f'=0$ or $f'$ 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.
Find all critical points of $f(x)=x^3-3x$.
$f'(x)=3(x-1)(x+1)=0$ at $x=\pm1$.
Every point where $f'(x)=0$ is a local maximum or minimum.
$y=x^3$ at $0$ has $f'=0$ but no extremum — a critical point is only a candidate.
On a closed interval, an absolute extremum must occur at a critical point or an...
Check critical points and the two endpoints.
$f'(c)=0$ does not guarantee an extremum at $c$. The curve $y=x^3$ has $f'(0)=0$, but $x=0$ is neither a max nor a min — just a flat inflection. A critical point is only a candidate; you must test it (lessons 5.4, 5.7). And the EVT needs both continuity and a closed interval.
Find the critical points of $f(x)=x^3-3x$.
- $f'(x)=3x^2-3=3(x^2-1)=3(x-1)(x+1)$.
- Set $f'(x)=0$: $x=1$ and $x=-1$ — both give a horizontal tangent.
- $f'$ is defined everywhere (a polynomial), so these two are the only critical points.
The Extreme Value Theorem: a function continuous on a closed interval attains an absolute max and min. Those, and all local extrema, occur only at critical points ($f'=0$ or undefined) or interval endpoints. A critical point is a candidate, not a guarantee — always test it.