Find local minimizers, local maximizers, global minimizers and global maximizers: (i) f(x, y) = x 2...

Question

Find local minimizers, local maximizers, global minimizers and global

maximizers:

(i) f(x, y) = x

2 − 4x + 2y

2 − 1; (ii) f(x, y, z) = 3x

2 + 2y

2 + z

2 + 2xy + 4xz +

yz; (iii) f(x, y, z) = xyz + 1 x + 1 y + 1 z, x, y, z ≠ 0

(iv) f(x, y) = (x + y)(xy + 1).

Baljit · Accepted Answer

(i) f(x, y) = x2 − 4x + 2y2 − 1; 
partially differentiate wrt x and y then put result  is equal to 0
=2x-4=0
=4y=0
From above equations we get x=2 and y=0.So stationary point is (2,0)
Now Calculate second order partial derivative at point (2,0)
r =
s=
t=
Now rt-s2=8-0=8
Now rt-s2>0 and r>0 so Given curve is minimum at point(2,0)
So (2,0) is minimizer .Since curve is minimum at only one point so this point is Global minimizer.
(ii)  f(x, y, z) = 3x2 + 2y2 + z2 + 2xy + 4xz +yz; 
partially differentiate wrt x ,

Find local minimizers, local maximizers, global minimizers and global maximizers: (i) f(x, y) = x 2 − 4x + 2y 2 − 1; (ii) f(x, y, z) = 3x 2 + 2y 2 + z 2 + 2xy + 4xz + yz; (iii) f(x, y, z) = xyz + 1 x...

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