Curve sketching, question 25

Topic: Curve sketching
Difficulty: Challenging
Suggested time: 7 min
Marks: 4
Theory: UBC Math 100 §7 CLP-1 §3.6Key results

A company sells $q$ units of a product a week, and must lower its price to sell more, so its weekly revenue in dollars is $R(q)$ below. Beyond the sales level that maximises revenue, revenue falls as $q$ grows. At what sales level is revenue falling fastest?

R(q) = 100q\,e^{-q/50}, \qquad q \ge 0

Give the number of units a week.

Your answer

units

A whole number.

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Theory UBC Math 100 §7 CLP-1 §3.6Key results

Worked solution

Strategy

The rate of change of revenue is R′. Its most negative value comes where R″ changes sign from negative to positive: the inflection point of R, not its maximum.

Working

$R'(q) = 100e^{-q/50} + 100q\cdot\left(-\tfrac{1}{50}\right)e^{-q/50} = e^{-q/50}(100 - 2q)$
Product rule; the chain rule gives −1/50 from e^−q/50. R′ = 0 at q = 50, the revenue-maximising level.
$R''(q) = -\tfrac{1}{50}e^{-q/50}(100 - 2q) + e^{-q/50}(-2) = e^{-q/50}\left(\frac{q}{25} - 4\right)$
Product rule again: −(100 − 2q)/50 − 2 = q/25 − 4.
$R''(q) < 0 \text{ for } q < 100, \qquad R''(q) > 0 \text{ for } q > 100$
So R′ decreases until q = 100 and increases after it: its smallest value is at q = 100.
$R'(100) = -100e^{-2} \approx -13.53$
Negative, so revenue is indeed falling there, faster than at any other sales level: about $13.53 per extra unit.

Answer

q = 100

Check

R′(90) ≈ −13.22, R′(100) ≈ −13.53 and R′(110) ≈ −13.30, so revenue is falling fastest at q = 100.

Where this goes wrong

Solving R′(q) = 0, which gives q = 50: the sales level of maximum revenue, where revenue is not falling at all. Falling fastest means R′ is most negative, which happens where R″ changes sign.

Leans on

The product rule, from Differentiation rules

Try a similar question

The number of fruit flies in a jar, in hundreds,

t

days after the jar is set up, is modelled by the logistic function below. At what time is the population growing fastest?

P(t) = \frac{10}{1 + 9e^{-t}}, \qquad t \ge 0

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Curve sketching practice question 25