MTH6131 Chap.10 Curve Sketching and Extreme Values
Curve Sketching and Extreme Values
The vocabulary of shape
The notes define each shape word separately. A function is increasing on an interval when larger inputs always give larger outputs, and decreasing when they give smaller ones. A stationary point has a zero derivative; a critical point is an interior point where the derivative is zero or does not exist, so every stationary point is critical but not the reverse.
An absolute maximum beats every value in the domain, while a relative maximum beats only nearby values.
The first and second derivative tests
The first derivative test finds all critical points, marks them on the axis, finds the sign of the derivative on each interval, and reads the changes: plus to minus is a relative maximum, minus to plus a relative minimum, no change is neither.
The second derivative test is quicker where it works: a negative second derivative at a stationary point means a maximum, a positive one a minimum.
When the second derivative is zero the test gives no conclusion, and the notes are explicit that it fails there, so the first derivative test takes over.
Concavity, inflection and asymptotes
A curve is concave up where every arc lies below its chord and concave down where every arc lies above it; with a second derivative, positive means concave up and negative means concave down.
An inflection point is where concavity changes, which needs a sign change of the second derivative, not just a zero.
Vertical asymptotes usually sit at zeros of a denominator, and horizontal ones come from limits at plus and minus infinity.
The sketching guidelines
The notes give nine steps: domain, symmetry, first and second derivatives, critical and inflection candidates, intervals of increase and concavity, extrema and inflection points, asymptotes and end behaviour, intercepts, and finally the sketch.
On a closed interval, absolute extrema come from comparing the values at the critical points and at the two ends.
What this chapter covers
- 01
Increasing, decreasing, stationary and critical
- 02
Relative against absolute extrema
- 03
The first derivative test and sign charts
- 04
The second derivative test and its silent case
- 05
Concavity and inflection points
- 06
Vertical and horizontal asymptotes
- 07
The nine sketching guidelines
- 08
Absolute extrema on a closed interval
Worked example · free
A quartic with two minima
- 2Differentiate and factor to find the critical points.
- 3Classify each critical point with the second derivative.
- 2Solve f″ = 0 and confirm the sign changes.
Key terms
- Critical Point
- An interior point of the domain where the derivative is zero or does not exist.
- Relative Maximum
- A point where the function value is at least as large as every value at nearby points.
- Concave Up
- Bending upward, with every arc of the curve lying below its chord; guaranteed where the second derivative is positive.
- Horizontal Asymptote
- A horizontal line that the curve approaches as the input tends to plus or minus infinity.
- Sign Chart
- A number line marked with the critical points and the sign of a derivative on each interval between them.
- Absolute Minimum
- The smallest value a function takes on its whole domain or on a stated interval.
Curve Sketching and Extreme Values FAQ
What should I do when the second derivative test gives zero?
Go back to the first derivative test. A zero second derivative at a stationary point means the test is silent: the point could be a maximum, a minimum or neither. The function x to the fourth has a minimum at zero, while x cubed has neither, and both have zero second derivative there.
Is every point where the second derivative is zero an inflection point?
No. An inflection point needs the concavity to change, which means the second derivative must change sign. The function x to the fourth has zero second derivative at the origin but is concave up on both sides, so the origin is not an inflection point. Inflection can also happen where the second derivative is undefined.
How do I find absolute extrema on a closed interval?
List the critical points inside the interval and the two endpoints, evaluate the function at every one, and pick the largest and smallest values. No sign test is needed, because the Extreme Value Theorem guarantees both extremes exist and they must occur at one of these candidates.
Where do vertical asymptotes come from?
Usually from zeros of a denominator that do not cancel with the numerator. At such a point the function's values grow without bound on at least one side. Justify each asymptote by writing the one-sided limits rather than by drawing, since a cancelled factor gives a hole instead.
Exam move
Sketch every practice function twice: once from the sign chart alone, before plotting anything, and once from computed points. Where the two disagree, find which sign was wrong. The sign chart is the part of a sketching answer that carries the reasoning, so it should be the part you trust.
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