- Difficulty
- Challenging
- Suggested time
- 6 min
- Marks
- 4
The chain rule and logarithms practice question 24
The curve has exactly two points where the tangent line is horizontal. Find the -coordinate of the higher of the two.
A horizontal tangent is where the slope vanishes. To find , notice that is a product and its exponential has , not , in the exponent.
Product rule, with the chain rule on the exponential: .
Factor out , which is never zero, and solve . Then compare the -values at the two solutions.
An exact value, or a decimal to three places.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Horizontal tangents are where y′ = 0. The exponential factor never vanishes, so after factoring it out the zeros come from a simple polynomial. Then evaluate y at each and take the larger.
Working
Product rule; the chain rule gives the inner derivative −2x/8 = −x/4.
The exponential is never zero, so only the bracket can vanish.
The two points mirror each other through the origin; the one at x = 2 lies above the axis.
The higher point is (2, 2/√e).
Answer
The answer is 2 over the square root of e, about 1.213.
Check
y(1.9) ≈ 1.2100 and y(2.1) ≈ 1.2101 both sit just below y(2) ≈ 1.2131, as values beside a peak should.
Where this goes wrong
Differentiating e−x²/8 as e−x²/8, without the inner derivative −x/4. Then y′ = e−x²/8(1 + x), which puts a false horizontal tangent at x = −1, where y = −e−1/8.
Leans on
The product rule, from Differentiation rules
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