- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 1
Curve sketching practice question 9
The graph of a function is shown with four labelled points. At which point are and both negative?
means the graph falls as you move to the right. means it bends downward, like an upside-down bowl.
The graph bends downward everywhere left of the origin and upward everywhere right of it. At each point on the downward-bending side, is the graph rising or falling?
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Read each derivative from the shape: the sign of f′ from whether the graph rises or falls, and the sign of f″ from which way it bends.
Working
Left of the origin the graph bends downward. It rises before the high point at x = −2 and falls after it.
Right of the origin it bends upward. It falls before the low point at x = 2 and rises after it.
Answer
The answer is At Q.
Check
Chords from the high point (−2, 2) to Q and on to the origin have slopes of about −0.6 and then about −1.4: falling, and more steeply.
Where this goes wrong
Mixing up falling with bending downward. P bends downward but is rising; R is falling but bends upward. Only at Q is the graph both falling and bending downward.
Leans on
Higher-order derivatives, from Differentiation rules
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