MAT2615
ASSIGNMENT 03
CORRECT SOLUTIONS
Due 2025
, Problem 1: Critical Points
Given f (x, y) = y ln x + 12 y 2 , find the critical points and classify them.
First-order partial derivatives:
∂f y ∂f
= , = ln x + y
∂x x ∂y
Set both to zero:
y
= 0 ⇒ y = 0, ln x + 0 = 0 ⇒ x = 1
x
Critical point: (1, 0)
Second-order partial derivatives:
y 1
fxx = − , fyy = 1, fxy = fyx =
x2 x
Hessian at (1, 0):
0 1
H= , det(H) = −1 < 0
1 1
Conclusion: Saddle point at (1, 0)
1
ASSIGNMENT 03
CORRECT SOLUTIONS
Due 2025
, Problem 1: Critical Points
Given f (x, y) = y ln x + 12 y 2 , find the critical points and classify them.
First-order partial derivatives:
∂f y ∂f
= , = ln x + y
∂x x ∂y
Set both to zero:
y
= 0 ⇒ y = 0, ln x + 0 = 0 ⇒ x = 1
x
Critical point: (1, 0)
Second-order partial derivatives:
y 1
fxx = − , fyy = 1, fxy = fyx =
x2 x
Hessian at (1, 0):
0 1
H= , det(H) = −1 < 0
1 1
Conclusion: Saddle point at (1, 0)
1
