5.9 Connecting a Function, Its First Derivative, and Its Second Derivative
- Syllabus
- 2020
- Topic
- 5.9
- Level
- —
Learning objectives
Match $f$, $f'$, and $f''$ by Aligned Features
At the same input x, the height f′(x) is the slope of the graph of f, and the height f′′(x) is the slope of the graph of f′. To match related graphs, compare features at aligned x-values; the graphs do not need to look alike.
| Feature on one graph | Aligned feature on the next graph |
|---|---|
| f increasing / decreasing | f′ above / below the x-axis |
| horizontal tangent on f | zero of f′ |
| f concave up / down | f′ increasing / decreasing |
| horizontal tangent on f′ | zero of f′′ |
| f′ increasing / decreasing | f′′ above / below the x-axis |
Start with unmistakable events such as zeros and horizontal tangents. Check the sign of the proposed derivative against increasing or decreasing intervals. Then check whether its own rise and fall agrees with the sign of the proposed second derivative. Require several consistent relationships before assigning labels.
For f(x)=x3−3x, f′(x)=3x2−3,f′′(x)=6x. The cubic has horizontal tangents at x=−1 and x=1, exactly where the derivative parabola crosses the x-axis. The parabola decreases for x<0 and increases for x>0, matching the negative and positive sides of the line f′′=6x. The cubic therefore changes concavity at x=0.
Do not match graphs by overall shape, height, or a single zero. A derivative records slope, not the original function's output. A proposed match is justified only when signs, zeros, turning behavior, and concavity agree at the same inputs.