Calculus I · 2B · lesson
Extrema and Critical Numbers
Learn extrema and critical numbers with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Section overview
Theorems, extrema, and curve shapeWhat this section is building
Learn extrema and critical numbers with clear exposition, guided derivations, worked examples, visuals, checks, and interpretation.
Critical numbers divide the domain into testable intervals; endpoints and discontinuities keep local evidence from becoming an unjustified global claim.
List the domain and candidates, test derivative signs, compare endpoint values, and verify each theorem's hypotheses explicitly.
Calling every point with f-prime zero an extremum or every point with f-double-prime zero an inflection point.
Learning objectives
Distinguish local and absolute extrema; locate critical numbers and endpoints that may contain extrema.
Derivative Theorems and the Shape of a Graph
Local and Absolute Extrema
Before the formulas
The theorem in Extrema and Critical Numbers connects global information on an interval with local derivative behavior inside it. Read the hypotheses and conclusion separately. The theorem guarantees existence of at least one point; it may not identify the point, make it unique, or place it at the midpoint.
Use a diagram to understand the claim, then return to algebra to find candidate values when the problem asks for them. A correct theorem citation should name the interval and explain why each hypothesis is satisfied.
Read this graph as text
Critical numbers are candidates, not automatic extrema. Horizontal tangents and nondifferentiable interior points create candidates. Endpoints matter on closed intervals. Function values must be compared to determine absolute extrema. The marked interior points are where f'=0 in this example. They must be evaluated, but not all are guaranteed to be maxima or minima. The endpoints are also candidates for absolute extrema on the closed interval. The final answer comes from comparing heights.
Every relationship in critical numbers are candidates, not automatic extrema is identified with written labels plus distinct solid, dashed, dotted, double, marker, or pattern cues; color is never the only carrier of meaning.
Why it matters: This graph should dismantle the false rule "set the derivative equal to zero and you have the maximum." It distinguishes candidate generation from classification and absolute comparison.
Horizontal tangents and nondifferentiable interior points create candidates. Endpoints matter on closed intervals. Function values must be compared to determine absolute extrema.
An extreme value can occur where the derivative is zero, undefined, or at an endpoint
A horizontal tangent is one possible signal of a local maximum or minimum, but it is not a guarantee. Flat points can also occur where the function continues through without turning. Critical numbers identify candidates that require further analysis.
Endpoints belong to absolute-extreme problems even though the ordinary two-sided derivative is not available there. Keeping local and absolute questions separate prevents missing the true largest or smallest value on a closed interval.
Optimization begins with vocabulary. A local maximum beats nearby values; an absolute maximum beats every feasible value. Critical numbers are inputs where or fails to exist, but they are candidates, not automatic extrema.
A function can pass through a critical number without turning, as does at the origin. The derivative identifies places worth investigating; additional sign or value analysis decides what actually happens.
A local maximum is larger than nearby function values. A local minimum is smaller than nearby values. An absolute maximum or minimum is largest or smallest on the entire domain or interval under consideration.
Critical number
A number in the domain of is critical when
or does not exist.
Critical numbers are candidates, not automatic extrema. Endpoints are also candidates for absolute extrema on closed intervals.
Fermat's theorem
If has a local extremum at an interior point and exists, then
Find critical numbers
Find the critical numbers of
Worked solution
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A critical number where the derivative fails
For , is critical because does not exist. It is also a local and absolute minimum. A search only for would miss it.
critical-01Find the critical numbers of .
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Critical numbers occur where f'=0 or where f' is undefined in the domain.
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