Math · Calculus I · Worked example
Sketch a cubic with its derivatives
Find where f(x) = x³ − 3x² − 9x + 5 increases and decreases, its local extrema, its concavity and its inflection point.
Find the critical points
f′(x) = 3x² − 6x − 9 = 3(x − 3)(x + 1), which is zero at x = −1 and x = 3.
Make a sign chart for f′
Test one number in each interval: f′(−2) = 15 > 0, f′(0) = −9 < 0 and f′(4) = 15 > 0.
| Interval | Test x | f′(x) | f is |
|---|---|---|---|
| x < −1 | −2 | 15 | increasing |
| −1 < x < 3 | 0 | −9 | decreasing |
| x > 3 | 4 | 15 | increasing |
Classify the critical points
f′ changes from + to − at x = −1, so f(−1) = 10 is a local maximum. It changes from − to + at x = 3, so f(3) = −22 is a local minimum.
Find where the concavity changes
f″(x) = 6x − 6 is negative for x < 1 and positive for x > 1, so the graph is concave down and then concave up. The inflection point is (1, f(1)) = (1, −6).
Check with the second derivative test
f″(−1) = −12 < 0 confirms the maximum, and f″(3) = 12 > 0 confirms the minimum.
Sketch the graph
Plot the maximum (−1, 10), the y-intercept (0, 5), the inflection point (1, −6) and the minimum (3, −22). The graph rises to the maximum, falls through the inflection point to the minimum, then rises again.
Result
f increases on (−∞, −1) and (3, ∞) and decreases on (−1, 3). Local maximum f(−1) = 10 and local minimum f(3) = −22. Concave down on (−∞, 1) and up on (1, ∞), with inflection point (1, −6).
Your turn
Find the local extrema and the inflection point of g(x) = x³ − 12x.
Show the answer and explanation
Local maximum g(−2) = 16, local minimum g(2) = −16, inflection point (0, 0).
g′(x) = 3x² − 12 = 3(x − 2)(x + 2) is zero at ±2, and g″(x) = 6x. g″(−2) = −12 < 0 gives the maximum g(−2) = 16; g″(2) = 12 > 0 gives the minimum g(2) = −16. g″ changes sign at 0, so (0, 0) is the inflection point.
Keep exploring
Open the Derivative Tracer and move the point from −1 to 3: the tangent is flat at both ends and steepest at the inflection point, x = 1, where the slope is −12.
Return to the concept →Sources and scope
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Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
Open the Derivative Tracer Graph f and its key points in Graph Check the derivatives in Math Open worked example on a board Derivative tests in Math ReferenceYour existing work stays on this device. Examples open as editable copies.