MAT2615
Assignment 3
Due June 2026
, Question 1
Problem Statement
Consider the function 𝑓: ℝ2 → ℝ defined by
𝑓(𝑥, 𝑦) = 𝑥 2 − 6𝑥 + 3𝑦 2 − 𝑦 3 .
(a) Find all the critical points of 𝑓.
(The function has two critical points.)
(b) Use Theorem 10.2.9 to determine the local extreme values and minimax values of 𝑓.
Also determine whether any of the local extrema are global extrema.
(a) Critical points
Step 1: Compute first-order partial derivatives
∂𝑓 ∂𝑓
𝑓𝑥 = ∂𝑥 = 2𝑥 − 6 𝑓𝑦 = ∂𝑦 = 6𝑦 − 3𝑦 2
Step 2: Set partial derivatives equal to zero
2
2𝑥 − 6 = 0 ⇒ 𝑥 = 3 6𝑦 − 3𝑦 =0
Factor:
3𝑦(2 − 𝑦) = 0
So:
𝑦=0 or 𝑦 = 2
Assignment 3
Due June 2026
, Question 1
Problem Statement
Consider the function 𝑓: ℝ2 → ℝ defined by
𝑓(𝑥, 𝑦) = 𝑥 2 − 6𝑥 + 3𝑦 2 − 𝑦 3 .
(a) Find all the critical points of 𝑓.
(The function has two critical points.)
(b) Use Theorem 10.2.9 to determine the local extreme values and minimax values of 𝑓.
Also determine whether any of the local extrema are global extrema.
(a) Critical points
Step 1: Compute first-order partial derivatives
∂𝑓 ∂𝑓
𝑓𝑥 = ∂𝑥 = 2𝑥 − 6 𝑓𝑦 = ∂𝑦 = 6𝑦 − 3𝑦 2
Step 2: Set partial derivatives equal to zero
2
2𝑥 − 6 = 0 ⇒ 𝑥 = 3 6𝑦 − 3𝑦 =0
Factor:
3𝑦(2 − 𝑦) = 0
So:
𝑦=0 or 𝑦 = 2
