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How to Identify Min/Max/Critical or Saddle Points?
Hello, currently I'm on my last attempt on the following question.
Consider the function
f(x,y) =(x^2)y +y^3 - 48y
f has __ at 4sqrt(3), 0)
f has __ at (0,0)
f has __ at (0, -4)
f has __ at (4sqrt(3), 0)
f has __ at (0,4)
I have tried (respectively)
Saddle Saddle
Critical No Critical Point
Minimum Minimum
Saddle Saddle
Maximum Maximum
The option are "Minimum, Maximum, Some Critical Point, No Critical Point, Saddle Point or ?"
Any help is greatly appreciated.
Thank you in advance!
1 Answer
- ?Lv 73 years ago
I’ll use the method described in this link (see blue ‘Fact’ box ¾-way down):
http://tutorial.math.lamar.edu/Classes/CalcIII/Rel...
f(x,y) =x²y + y³ – 48y
∂f/∂x = 2xy
∂²f/∂x² = 2y
∂f/∂y = x² + 3y² – 48
∂²f/∂y² = 6y
∂²f/(∂x∂y) = 2x
D = (∂²f/∂x²)(∂²f/∂y²) - (∂²f/(∂x∂y))² = (2y)(6y) – (2x)² = 12y² - 4x²
_______________________
Identify the critical points: if ∂f/∂x=0 and ∂f/∂y=0, this is a critical point.
For each critical point, find D and classify point (using D and ∂²f/∂x²) as a rel.max./rel.min//saddle.
_______________________
Point A(4√3, 0)
∂f/∂x = 2xy = 2*4√3*0 = 0
∂f/∂y = x² + 3y² – 48 = (4√3)² + 3(0)² – 48 = 0
So this is a critical point.
D = 12y² – 4x² = 12(0)² – 4(4√3)² = negative value. D<0
So (4√3, 0) is a saddle point.
_______________________
Point B(0,0)
∂f/∂x = 2xy = 2*0*0 = 0
∂f/∂y = x² + 3y² – 48 = (0)² + 3(0)² – 48 = negative
So this is a not a critical point.
_______________________
Point C(0, -4)
∂f/∂x = 2xy = 2*0*(-4) = 0
∂f/∂y = x² + 3y² – 48 = (0)² + 3(-4)² – 48 = 0
So this is a critical point.
D = 12y² – 4x² = 12(-4)² – 4(0)² = positive value. D>0
∂²f/∂x² = 2y = 2*(-4) = negative
So this is a relative maximum.
_______________________
Point D(4√3, 0)
This is the same as point A. Maybe you have a typing error.
_______________________
Point E(0, 4)
∂f/∂x = 2xy = 2*0*(4) = 0
∂f/∂y = x² + 3y² – 48 = (0)² + 3(4)² – 48 = 0
So this is a critical point.
D = 12y² – 4x² = 12(4)² – 4(0)² = positive value. D>0
∂²f/∂x² = 2y = 2*4 = positive
So this is a relative minimum.
_______________________
Summary:
Critical: saddle
Not critical
Critical: relative maximum
Probably error as point is same as 1st point
Critical: relative minimum