- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
Curve sketching practice question 20
The graph of the derivative of a function is shown: it is the graph of , not of . How many inflection points does the graph of have?
An inflection point of is where changes sign, and is the derivative of : the slope of the drawn curve.
The slope of the drawn curve changes sign where switches between rising and falling, at its high and low points.
A whole number.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
f″ is the slope of the drawn graph. Count the places where that slope changes sign, which are the turning points of f′.
Working
Reading the drawn graph from left to right.
f″ is the slope of the graph of f′, so it has the sign of each rise or fall.
One sign change at each turning point of the drawn graph: three inflection points of f.
Answer
The answer is 3.
Check
f′ is zero at −2, −1, 1 and 2, and between each pair of neighbouring zeros it must turn, so it has at least three turning points; the graph shows exactly three, each an inflection point of f.
Where this goes wrong
Counting where the drawn graph crosses the x-axis. Those four points are where f′ = 0, the local extrema of f; inflection points of f come from the turning points of f′.
Leans on
Higher-order derivatives, from Differentiation rules
Try a similar question
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