Short on time? This page is the recap. For explanations, figures and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- Information about f can come as a formula, a graph of f′ or a table of values. The rules are the same for all three.
- Sign-chart method: find where f′ and f″ are zero or undefined, put these on one number line, test signs in each interval, calculate f at each marked point, then sketch.
- Reading a graph of f′: above or below the axis tells you whether f rises or falls; going up or going down tells you whether f is concave up or down.
- A peak or trough on the graph of f′ is a point of inflection of f, not an extremum.
- Reverse direction: horizontal tangents of f are zeros of f′; inflection points of f are peaks or troughs of f′.
Key relationships
| f′ | f″ | Shape of f |
|---|---|---|
| + | + | rising, getting steeper |
| + | − | rising, flattening out |
| − | + | falling, flattening out |
| − | − | falling, getting steeper |
| Feature of f | Evidence |
|---|---|
| relative maximum | f′ changes from + to − |
| relative minimum | f′ changes from − to + |
| point of inflection | f″ changes sign, or f′ has a peak or trough |
| horizontal tangent, no extremum | f′ = 0 but keeps the same sign on both sides |
Assumptions behind the method
- f is continuous on each interval you sketch.
- Signs of f′ and f″ only change at the points you marked (zeros, undefined points).
- With a table, you need extra information about f′ between the listed values.
Mistakes to avoid
- Treating the graph of f′ as the graph of f.
- Assuming every zero of f′ is an extremum, or every zero of f″ an inflection point.
- Using the sign of f′ for concavity. Concavity comes from whether f′ is increasing or decreasing.
- Straight-line joins between key points instead of the correct bend.
- Unlabelled sketches. Give coordinates of every extremum and inflection point.
Quick self-check
- Let r(x) = x³ − 12x. Give the relative extrema and the inflection point. (r′ = 3(x − 2)(x + 2), r″ = 6x. Relative maximum (−2, 16), relative minimum (2, −16), inflection (0, 0).)
- On an interval, f′ is negative and increasing. Describe f. (Falling and concave up: falling, flattening out.)
- The graph of f′ touches the x-axis at x = 5 without crossing it. Is there an extremum of f at x = 5? (No. f′ keeps the same sign on both sides. But f′ has a peak or trough there, so f has a point of inflection with a horizontal tangent.)
Next: practice questions.