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