lOMoARcPSD|44389878
, lOMoARcPSD|44389878
MAT2615 Assignment 3 2026 - DUE June 2026 [COMPLETE ANSWERS]
Question 1
Function:
f ( x , y )=x 2−6 x +3 y 2− y 3
(a) Find all critical points of f
Critical points occur when the gradient ∇ f (x , y)=0. Compute the partial derivatives:
∂f ∂f 2
f x= =2 x−6 f y = =6 y−3 y
∂x ∂y
Set each partial derivative to zero:
1. f x =0 ⟹ 2 x−6=0 ⟹ x=3
2. f y =0 ⟹ 6 y −3 y 2=0 ⟹ 3 y (2− y )=0 ⟹ y =0 or y=2
✅ So the critical points are:
¿
(b) Classify the critical points using Theorem 10.2.9 (Second Derivative Test)
Compute the second partial derivatives:
∂2 f ∂2 f 2
∂ f
f xx = 2
=2f yy = 2
=6−6 y f xy = =0
∂x ∂y ∂x∂ y
The Hessian determinant D( x , y)is:
D( x , y)=f xx f yy −¿
Check each critical point
1. At ( 3,0 ) :
D=12−12(0)=12>0 , f xx =2> 0
✅ Local minimum.
2
f (3,0)=3 −6(3)+3 ¿
, lOMoARcPSD|44389878
MAT2615 Assignment 3 2026 - DUE June 2026 [COMPLETE ANSWERS]
Question 1
Function:
f ( x , y )=x 2−6 x +3 y 2− y 3
(a) Find all critical points of f
Critical points occur when the gradient ∇ f (x , y)=0. Compute the partial derivatives:
∂f ∂f 2
f x= =2 x−6 f y = =6 y−3 y
∂x ∂y
Set each partial derivative to zero:
1. f x =0 ⟹ 2 x−6=0 ⟹ x=3
2. f y =0 ⟹ 6 y −3 y 2=0 ⟹ 3 y (2− y )=0 ⟹ y =0 or y=2
✅ So the critical points are:
¿
(b) Classify the critical points using Theorem 10.2.9 (Second Derivative Test)
Compute the second partial derivatives:
∂2 f ∂2 f 2
∂ f
f xx = 2
=2f yy = 2
=6−6 y f xy = =0
∂x ∂y ∂x∂ y
The Hessian determinant D( x , y)is:
D( x , y)=f xx f yy −¿
Check each critical point
1. At ( 3,0 ) :
D=12−12(0)=12>0 , f xx =2> 0
✅ Local minimum.
2
f (3,0)=3 −6(3)+3 ¿
