- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
The chain rule and logarithms practice question 28
The curve has exactly two points with a horizontal tangent. One of them is the origin. Find the -coordinate of the other.
The origin already gives one horizontal tangent. Once you have , which of its factors can be zero, and which never can?
Differentiate with the product rule and the chain rule, then factor out , which is never zero.
. Set it to zero and find the non-zero root, then evaluate there.
An exact expression, or a decimal to two decimal places.
Theory UBC Math 100 §4.3CLP-1 §2.9Key results
Worked solution
Strategy
Horizontal tangents are zeros of the derivative. The exponential factor never vanishes, so the zeros come from the polynomial factor; one is x = 0 and the other is the one wanted.
Working
Product rule; the chain rule gives the −1 in the derivative of e−x.
e−x > 0 always, so only the polynomial factor can vanish.
The y-coordinate at the other point.
Answer
The answer is 4 over e squared, about 0.541.
Check
Beside x = 2 the curve is lower on both sides, y(1.9) ≈ 0.5399 and y(2.1) ≈ 0.5400 against 0.5413 at the peak.
Where this goes wrong
Losing the minus sign from the derivative of e−x, which gives y′ = e−x(2x + x²) and a horizontal tangent at x = −2 instead. That point is on the curve, but its tangent is not horizontal.
Leans on
The product rule, from Differentiation rules
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