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Find the critical points and classify them as local maxima, local minima, saddle points, or none of

Author :

Submitted : 2018-06-14 06:05:39    Popularity:     

Tags: points  classify  Find  critical  minima  

f(x,y) = (x^2)y + (2y^2) - 6xy + 1

Answers:

f(x,y) = (x^2)y + (2y^2) - 6xy + 1
∂f/∂x = f_x = 2xy-6y = 2y(x-3) = 0
∂f/∂y = f_y = x^2-6x+4y = 0
Points are
(0,0), (3,9/4) and (6,0)
∂^2f/∂x^2 = f_xx = 2y
∂^2f/∂y^2 = f_yy = 4
f_xy = 2x-6
H = f_xx*f_yy - (f_xy)^2
At (0,0)
H = 0-36 = -36 (Saddle Point)
At (6,0)
H = 0-36 = -36 (Saddle Point)
At (3, 9/4)
H = (9/2)*4 - 9 = 9 (Minimum)

Per WolframAlpha:
Definition:
A point of a function's graph where the 1st derivative is either zero or undefined.

f_x = 2 xy - 6y = 2 y ( x - 6) ; f_y = x² + 4 y -6 x..----> f_y = 0 means 4y = 6 x - x²---> f_x = 0 means either y = 0 or x = 6...

points ( 0 , 0 ) , ( 6 , 0 ) ; ...f_xx = 2y ; f_yy = 4 ; f_xy = 2x - 6...both are saddles

First, can you sate here whats the procedure to find/classify the critical points?
State it here and we can then take it from there,,,



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