MATH1011 Chap.10 Extrema, Concavity and Curve Sketching
Extrema, Concavity and Curve Sketching
Stationary points are candidates, not verdicts
A stationary point occurs where the first derivative is zero, but that fact alone does not tell you whether the graph has a maximum, a minimum or neither. The derivative records direction. A change from positive to negative gives a local maximum, a change from negative to positive gives a local minimum, and no sign change gives neither.
Build a sign chart from the critical numbers rather than treating every solution of f prime equals zero as an extremum.
Critical numbers also include interior domain points where the derivative fails to exist. Endpoints require separate treatment because a closed interval can achieve an absolute maximum or minimum there even though a two-sided derivative is unavailable.
For an absolute-extrema problem on a closed interval, evaluate every interior critical point and both endpoints in the original function, then compare the function values.
Concavity explains how the slope itself changes
The second derivative measures the change in slope.
Where it is positive, slopes are increasing and the graph is concave up; where it is negative, slopes are decreasing and the graph is concave down. An inflection point requires an actual change in concavity.
Solving f double prime equals zero or finding where it is undefined only produces candidates, just as the first derivative produces candidates for extrema.
The second derivative test is a shortcut at a stationary point. A positive second derivative identifies a local minimum and a negative one identifies a local maximum. A zero second derivative is inconclusive, not a declaration that no extremum exists.
Return to the first-derivative sign test when the shortcut gives no verdict.
A curve sketch is a reconciliation of several facts
Start with the domain, intercepts and any symmetry. Add vertical, horizontal or oblique asymptotes from the limit analysis. Use the first derivative to mark increasing and decreasing intervals, then the second derivative to mark concavity and inflection.
The finished curve must respect every item simultaneously: it cannot rise through an interval labelled decreasing or cross a vertical asymptote.
Optimisation problems use the same analysis after one modelling step. Express the quantity to be optimised as a function of one variable, identify the feasible interval from the geometry or economics of the situation, and then compare legitimate candidates.
A stationary value outside that feasible interval is not a solution, and an endpoint can win even when an interior stationary point exists.
What this chapter covers
- 01
Critical numbers and stationary points
- 02
First-derivative sign changes
- 03
Absolute extrema on closed intervals
- 04
Concavity and the second derivative
- 05
Inflection points as changes in concavity
- 06
Optimisation with a feasible domain
Worked example · free
Classify stationary points and compare absolute values
- 2Differentiate and find all interior critical numbers.
- 3Use derivative signs or the second derivative to classify the stationary points.
- 3Evaluate the endpoints and compare all candidate values.
Key terms
- Critical Number
- A Critical Number is an interior domain value where the derivative is zero or does not exist.
- Stationary Point
- A Stationary Point is a point on the graph where the first derivative equals zero.
- Local Extremum
- A Local Extremum is a maximum or minimum relative to sufficiently nearby domain points.
- Concavity
- Concavity describes whether slopes increase or decrease as the input moves from left to right.
- Inflection Point
- An Inflection Point is a domain point across which the graph changes concavity.
Extrema, Concavity and Curve Sketching FAQ
Does f prime equal zero always give a maximum or minimum?
No. It gives a stationary candidate. Classify it by checking how the first derivative changes sign, or use the second derivative when that test is conclusive. A horizontal tangent can occur without either kind of extremum if the function keeps increasing or keeps decreasing through the point.
Why do endpoints matter in an absolute-extrema problem?
An absolute maximum or minimum compares every allowed value, and closed endpoints belong to the domain. They may achieve the largest or smallest function value even though a two-sided derivative is unavailable there. Evaluate them alongside all interior critical points.
Is every solution of f double prime equals zero an inflection point?
No. It is only a candidate. Concavity must actually change from up to down or down to up across that input, and the point must belong to the function's domain. A zero second derivative with the same sign on both sides produces no inflection.
Exam move
Use one four-row routine on every curve: domain and asymptotes, critical numbers, first-derivative signs, second-derivative signs. Sketch only after the table is complete. For optimisation, write the feasible interval beside the objective function before differentiating, then reject any stationary candidate outside it and compare the remaining values with the endpoints.
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