- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Curve sketching practice question 10
On which interval is concave down?
Concavity is governed by the sign of the second derivative, not the first.
, so . Factor it and find where it is negative.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Differentiate twice, find where the second derivative is zero, and test the sign on each resulting interval. Concave down means the second derivative is negative.
Working
First derivative. Not what determines concavity, but needed to get to the second.
Second derivative, factored so the sign changes are visible.
(x − 1)(x + 1) is negative exactly when its two factors have opposite signs, which is between −1 and 1.
The points x = ±1 are inflection points, where the concavity changes.
Answer
The answer is the interval from −1 to 1.
Check
f(0) = 0 sits above the chord from x = −0.5 to 0.5, at −1.4375, but f(1.5) = −8.4375 sits below the chord from f(1) = −5 to f(2) = −8.
Where this goes wrong
Solving f′(x) = 0 instead of f″(x) = 0, which gives x = 0 and x = ±√3 and leads to the interval (−√3, √3). The first derivative describes rising and falling; the second describes bending.
Leans on
Sign charts, from Functions, graphs and inverses
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