- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Curve sketching practice question 2
Find the -coordinate of the inflection point of .
Is an inflection point about where turns, or about where it changes the way it bends? Which derivative measures that?
. Differentiate once more and find where changes sign.
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Concavity is governed by the second derivative. For a cubic, f″ is linear, so it changes sign exactly once, at its zero.
Working
Two derivatives.
Concave down then concave up: a genuine inflection point at x = 2.
Answer
The answer is 2.
Check
The roots of x³ − 6x² + x add to 6, and the inflection point of a cubic sits at the average of its three roots, 6/3 = 2.
Where this goes wrong
Setting f′ = 0 instead of f″ = 0, which finds the local extrema rather than the inflection point.
Leans on
Higher-order derivatives, from Differentiation rules
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