Topic 5.2 Extreme Value Theorem, Global Versus Local Extrema, and Critical Points: identify critical points and distinguish local from global extrema.

What this topic is asking

The College Board (Topic 5.2) sets up the language of optimization. You must state the Extreme Value Theorem (EVT), find critical points (where $f'(x) = 0$ or is undefined), and distinguish local (relative) extrema from global (absolute) extrema.

The Extreme Value Theorem

Both conditions are essential: the interval must be closed (endpoints included) and $f$ must be continuous. On an open interval, or with a discontinuity, an extremum can fail to exist (for example $f(x) = x$ on the open interval $(0, 1)$ attains neither a maximum nor a minimum).

Critical points

Local versus global extrema

A local maximum is a value that is at least as large as the function values nearby; a global (absolute) maximum is the largest value over the entire interval. The two can coincide, but they need not: a function can have several local maxima with only one of them being the global maximum. On a closed interval the global extrema are found among the critical points and the endpoints, an idea formalised by the candidates test.

Why "in the domain" matters

A point can only be a critical point if it is in the domain of $f$. For $f(x) = \frac{1}{x}$, the derivative $f'(x) = -\frac{1}{x^2}$ is undefined at $x = 0$, but $x = 0$ is not in the domain of $f$, so it is not a critical point. The same caution applies to functions with restricted domains: only points where $f$ itself is defined count. Exam questions exploit this by giving functions with cusps, holes, or roots, where the derivative is undefined at a point that may or may not be in the domain. Always confirm the point belongs to the domain before calling it critical.

How the EVT and critical points work together

The Extreme Value Theorem and the idea of critical points are two halves of the same optimization story. The EVT guarantees that a continuous function on a closed interval actually has an absolute maximum and minimum, so the search is not in vain. Critical points (plus the endpoints) then tell you where to look: a theorem ensures that any interior extremum must occur at a critical point, so the only candidates are the critical points and the two endpoints. This is exactly the logic that the candidates test of the next topic formalises. Without the EVT you would not know an extremum exists; without critical points you would not know where it could be. Together they reduce an infinite search over the interval to checking a short, finite list of candidate $x$-values.

Verbal cues that signal each tool

Exam wording tells you which idea to invoke. A phrase like "explain why $f$ must have an absolute maximum on $[a, b]$" is asking for the Extreme Value Theorem, so the justification states that $f$ is continuous on the closed interval and cites the theorem by name. A phrase like "find all critical points" is asking you to solve $f'(x) = 0$ and find where $f'$ is undefined within the domain. A request to "find the absolute maximum value" combines both: you confirm existence (often implicitly) and then evaluate the candidates. Mapping the question's verb to the right tool, existence versus location versus value, keeps your response aimed at the marks the grader is looking for.