- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Optimization in two variables and Lagrange multipliers practice question 1
The function has exactly one critical point, . Find .
A critical point needs both partial derivatives to be zero at once. Which term of involves both variables, and what does it contribute to each partial?
When differentiating with respect to , is held fixed, so gives ; with respect to , it gives . Set both partial derivatives to zero and solve the two equations together.
A whole number.
Theory UBC Math 100 §16.1Key results
Worked solution
Strategy
Both partial derivatives exist everywhere, so the critical points are where both are zero. Differentiate with respect to each variable, holding the other fixed, and solve the two equations together.
Working
With y held fixed, xy differentiates to y; with x held fixed, to x.
Solve f_x = 0 for y and substitute into f_y = 0.
Back-substitute. Check: f_x = 4 + 1 − 5 = 0 and f_y = 2 + 2 − 4 = 0.
The sum asked for.
Answer
The answer is 3, from the critical point (2, 1).
Check
f(2, 1) = −7, while each neighbour (1, 1), (3, 1), (2, 0) and (2, 2) gives −6, so (2, 1) sits at the bottom of the bowl.
Where this goes wrong
Leaving out the xy term when differentiating, so f_x = 2x − 5 and f_y = 2y − 4. That gives (5/2, 2) and a sum of 4.5. With y held fixed, xy is a constant times x and differentiates to y.
Leans on
Partial derivatives, from Functions of two variables and partial derivatives
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