Short on time? This page is the recap. For explanations, the graphs and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Recap
- f″ is the slope of f′, and f′ is the slope of f. Every connection comes from this chain.
- On a graph of f′, look at two things: its sign (above or below the axis) and its direction (rising or falling).
- Sign of f′ → f increasing or decreasing. Direction of f′ → f concave up or down.
- Sign of f″ → f concave up or down and f′ increasing or decreasing. It does not tell you whether f rises or falls.
- The graph of f′ never gives values of f, only how f changes.
Key relationships
| On f | On f′ | On f″ |
|---|---|---|
| Increasing | Above axis | No information |
| Decreasing | Below axis | No information |
| Relative max | Crosses from above to below | Often negative (not required) |
| Relative min | Crosses from below to above | Often positive (not required) |
| Horizontal tangent, no extremum | Touches axis, no sign change | No information alone |
| Concave up | Rising | Above axis |
| Concave down | Falling | Below axis |
| Inflection point | Peak or valley | Changes sign |
Matching three graphs: zeros of one curve line up with turning points of the curve one level up. Check signs too, then build the chain f → f′ → f″.
Assumptions behind the rules
- f is continuous at any point you call an extremum or inflection point.
- “Concave up because f′ is increasing” needs f′ to be increasing on an open interval, not just at a point.
- An inflection point can occur where f″ does not exist (a corner on the graph of f′), as long as concavity changes.
Mistakes to avoid
- “f′ decreasing, so f decreasing.” That is concavity. Direction of f needs the sign of f′.
- Using zeros of f′ as inflection points. Inflection points come from turning points of f′.
- “f″(c) = 0, so inflection point.” f″ must change sign.
- Concluding “f is increasing” from a graph of f″.
- Reasons with no function named, such as “the graph turns”. Write “f′ changes from positive to negative at x = c”.
- Reading f(c) from the height of f′.
Quick self-check
- On an interval, f′ is negative and increasing. Describe f. (Decreasing and concave up.)
- f′(x) = x³ − 6x² = x²(x − 6). Where does f have a relative extremum? (Only at x = 6, a minimum: f′ changes from negative to positive there. At x = 0, f′ touches zero without changing sign.)
- For the same f′, where are the inflection points of f? (f″(x) = 3x(x − 4), which changes sign at x = 0 and x = 4.)
- A graph of f″ is positive on (2, 7). Can you say f is increasing on (2, 7)? (No. You only know f is concave up and f′ is increasing there.)
Next: practice questions.