- Topic
- Curve sketching
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 3
Curve sketching practice question 13
Find the -coordinate of the local maximum of .
At a local maximum the slope is zero. is a product with a composition inside it: which rules does its derivative need?
Product rule and chain rule: . The exponential never vanishes.
Check the sign of on either side of the critical point to confirm it is a maximum.
A number, as a fraction or a decimal.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Differentiate, factor out the positive exponential, find the single critical point and confirm it is a maximum from the sign change of f′.
Working
Product rule; chain rule on e−2x.
e−2x > 0 always.
Rising then falling: a local maximum, with value f(1/2) = 1/(2e).
Answer
The answer is one half.
Check
f(0.4) ≈ 0.180 and f(0.6) ≈ 0.181 both fall short of f(0.5) ≈ 0.184.
Where this goes wrong
Losing the factor −2 from the derivative of e−2x, which gives f′ = e−2x(1 + x) and a critical point at x = −1 that is not really there.
Leans on
The product rule, from Differentiation rules
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