Mistake Master
Home› Unit 5 · Analytical Applications of Differentiation› 5.1·5.2·5.3·5.4·5.5·5.6·5.7·5.8·5.9·5.10·5.11·5.12

Extreme Value Theorem, Global Versus Local Extrema, and Critical Points AB & BC

The Extreme Value Theorem guarantees that a function continuous on a closed, bounded interval attains both an absolute maximum and an absolute minimum on it. All three conditions are needed, and each fails on its own: $f(x) = x$ on the open $(0, 1)$ attains neither, a quadratic on $[1, \infty)$ has no maximum, and a function with a single reassigned value at $x = 2$ climbs toward a maximum it never reaches. When a condition fails the theorem is silent, which is weaker than saying no extremum exists.

A critical point is a number $c$ in the domain of $f$ where $f'(c) = 0$ or $f'(c)$ fails to exist. Searching only the first clause misses the cusp of $x^{2/3}$ at $0$, which is that function's absolute minimum; ignoring the domain rule wrongly promotes the asymptote of $\frac{x}{x-2}$ at $x = 2$, where $f$ has no value at all. Every interior extremum is a critical point, but the converse is false: $x^{3}$ and $x^{1/3}$ are both critical at $0$ and neither turns around. What decides the question is whether $f'$ changes sign.

THE EXTREME VALUE THEOREM NEEDS ALL THREE WORDS: CONTINUOUS, CLOSED, BOUNDED. [0, 1] CLOSED (0, 1) OPEN [0, 1], f(1) = 0.5 CONTINUOUS: YES CONTINUOUS: YES CONTINUOUS: NO MAX = 1, MIN = 0 INTERVAL NOT CLOSED MIN = 0 ATTAINED BOTH ATTAINED NEITHER ATTAINED NO MAXIMUM EVT SAYS BOTH EXTREMA EXIST. IT NEVER SAYS WHERE, AND WHEN A CONDITION IS MISSING IT SAYS NOTHING AT ALL, WHICH IS NOT THE SAME AS SAYING NOTHING EXISTS.
Hollow circles mark values the function approaches without reaching. The right panel is closed and bounded and still has no maximum.
FOUR CRITICAL POINTS. THE TANGENT BEHAVIOUR DOES NOT DECIDE WHICH ARE EXTREMA. f = 4 − x² f = x³ f = |x| f = x^(1/3) f′ = 0 f′ = 0 f′ UNDEFINED f′ UNDEFINED CRITICAL: YES CRITICAL: YES CRITICAL: YES CRITICAL: YES MAXIMUM NOT AN EXTREMUM MINIMUM NOT AN EXTREMUM TWO OF THE FOUR TURN AROUND. WHAT SEPARATES THEM IS NOT WHETHER f′ VANISHED OR FAILED, BUT WHETHER f′ CHANGED SIGN. A CRITICAL POINT IS A CANDIDATE.
One column per critical point. The two red dots are places where the derivative does something dramatic and the function does not turn.

The work

3 ways in · any order
Lesson
Extreme Value Theorem, Global Versus Local Extrema, and Critical Points ›

Checks the three conditions the Extreme Value Theorem needs and watches each fail on its own, separates relative from absolute extrema and value from location, defines a critical point as both kinds of derivative failure subject to the domain rule, and shows the one-way street: every interior extremum is critical, and most critical points are not extrema.

Skill check · 10 scenarios
Open lesson →
Diagnostic
10-item topic check ›

Ten items spanning the three failure modes of this topic: asserting that extrema exist without continuity on a closed and bounded interval, treating a horizontal or missing tangent as a turning point, and searching only where the derivative equals zero while skipping corners, cusps, and the domain rule.

Not yet available · 10 items
Coming soon
Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

Take the diagnostic to identify your misconceptions
Diagnostic first