Calculus AB · Unit 5: Analytical Applications of Differentiation · 14 min read · Updated 2026-05-10
Extreme Value Theorem, Global vs Local Extrema, Critical Points — AP Calculus AB
AP Calculus AB · Unit 5: Analytical Applications of Differentiation · 14 min read
1. The Extreme Value Theorem★★☆☆☆⏱ 3 min
The Extreme Value Theorem (EVT) is an existence theorem: it tells you when you are guaranteed to find a global maximum and global minimum, but it does not tell you how to find them.
Intuition: If you draw a continuous curve from $(a, f(a))$ to $(b, f(b))$ without lifting your pen, the curve must reach a highest point and a lowest point. The guarantee breaks if either condition fails: open intervals let you approach an extremum without reaching it, and discontinuities (like vertical asymptotes) let the function grow without bound.
Exam tip: Always state both EVT conditions for full justification points
2. Critical Points★★★☆☆⏱ 4 min
By Fermat's Theorem, if $f(x)$ has a local extremum at an interior point $c$, then $c$ must be a critical point. The converse is *not* true: not all critical points are local extrema.
3. Local vs Global (Absolute) Extrema★★★☆☆⏱ 4 min
Extrema are classified by the interval over which they are the maximum or minimum. The key distinctions tested on the AP exam are:
**Global (Absolute) Extremum**: A global maximum on an interval $I$ is a value $f(c)$ such that $f(c) \geq f(x)$ for *all* $x$ in $I$. A global minimum follows the reverse inequality. Global extrema can occur at critical points *or* endpoints. There can be only one global maximum value and one global minimum value per interval, though the same value can occur at multiple $x$-locations.
**Local (Relative) Extremum**: A local maximum at $c$ is a value $f(c)$ such that $f(c) \geq f(x)$ for all $x$ in some small open interval around $c$. Local extrema only occur at interior points, so endpoints can never be local extrema. Any global extremum at an interior point is always also a local extremum.
4. AP-Style Worked Practice★★★★☆⏱ 3 min
Common Pitfalls
Why: Students remember EVT requires continuity but forget the interval must also be closed and bounded
Why: Many introductory resources simplify definitions, but AP CED defines critical points as interior points only
Why: Students stop at checking the derivative and forget to confirm the point is in the original function's domain
Why: Students reverse Fermat's Theorem, incorrectly assuming all critical points must be extrema. For example, $f(x)=x^3$ has a critical point at $x=0$ that is not an extremum
Why: Students only check critical points and assume the global extremum is interior
Why: Students confuse the location ($x$-value) with the value of the extremum ($y$-value)