Calculus BC · CED Unit 5: Analytical Applications of Differentiation · 14 min read · Updated 2026-05-11
Extreme Value Theorem, Global vs Local Extrema, Critical Points — AP Calculus BC
AP Calculus BC · CED Unit 5: Analytical Applications of Differentiation · 14 min read
1. Critical Points★★☆☆☆⏱ 4 min
This definition follows from Fermat's Theorem, which states that if a function $f$ has a local extremum at an interior point $c$, then $c$ must be a critical point. A common misconception is that all critical points are extrema; this is not true: $f(x) = x^3$ has a critical point at $x=0$, but it is not an extremum.
2. The Extreme Value Theorem★★☆☆☆⏱ 3 min
If either condition fails, EVT does not guarantee global extrema exist. They may exist by chance, but you cannot rely on the theorem to confirm their existence. For example, $f(x) = 1/x$ on the open interval $(0, 1)$ is continuous but has no global maximum.
3. Local vs Global (Absolute) Extrema★★★☆☆⏱ 4 min
Extrema are classified by the interval over which they are the maximum or minimum value. The AP exam frequently tests the key distinctions between these two classifications:
**Global (Absolute) Maximum**: A value $f(c)$ such that $f(c) \geq f(x)$ for *all* $x$ in the interval $I$.
**Global (Absolute) Minimum**: A value $f(c)$ such that $f(c) \leq f(x)$ for *all* $x$ in the interval $I$.
**Local (Relative) Maximum**: A value $f(c)$ such that $f(c) \geq f(x)$ for all $x$ in some small open interval around $c$ (only nearby points, not the entire interval).
**Local (Relative) Minimum**: A value $f(c)$ such that $f(c) \leq f(x)$ for all $x$ in some small open interval around $c$.
Key AP-tested distinctions:
Global extrema can occur at critical points *or endpoints*, but local extrema only occur at interior critical points (endpoints cannot be local extrema, since you cannot have an open interval around the endpoint within the domain).
A function can have multiple local extrema, but only one global maximum value and one global minimum value (though these values can occur at multiple points).
Any global extremum at an interior point is automatically a local extremum, but global extrema at endpoints are never local.
4. AP-Style Practice Worked Examples★★★☆☆⏱ 5 min
Common Pitfalls
Why: Students confuse the requirement to check endpoints for global extrema with the definition of a critical point, which requires the point to be interior to the interval.
Why: Students memorize 'EVT gives extrema' but forget that both conditions are required to apply the theorem.
Why: Most introductory examples use polynomials that are differentiable everywhere, so students forget the second half of the critical point definition.
Why: Students assume all global extrema are automatically local, but the definition of local extrema requires an open interval around the point within the domain.
Why: Students misremember Fermat’s Theorem as working both directions, when Fermat’s only says all local extrema are at critical points, not the reverse.