Answer to Question #120421 in Operations Research for Madelyn Beahan

Question #120421
Optimize{z=f(x,y)=6x^2-9x-3xy-7y+5y^2 }
using the step-by-step procedure.
1
Expert's answer
2020-06-15T14:40:00-0400

Equation given is


"f(x,y) = 6x^{2} - 9x-3xy-7y+5y^{2}"


taking partial derivative with respect to x and y


"\\frac{\\partial f}{\\partial x} = 12x-9-3y"


"\\frac{\\partial f}{\\partial y} = -3x-7+10y"



solving above equation to find x and y


"12x-9-3y=0" "\\implies y = 4x-3" . . . . . . . (i)


"-3x-7+10y=0" . . . . . . . (ii)


putting value of y from (i) to (ii),

"-3x-7+10(4x-3) = 0"


"-3x-7 + 40x -30 = 0 \\implies x=1"


then from (i),

"y=4(1) - 3 = 1"


taking higher partial derivatives,

"r =\\frac{\\partial^{2} f}{\\partial x^{2} } = 12"


"t= \\frac{\\partial^{2} f}{\\partial y^{2} } = 10"


"s= \\frac{\\partial^{2} f}{\\partial x \\partial y } = -3"

Now

"rt-s^{2} = 12*10 - 9 > 0"


So point (1,1) is point of local minima.


Minimum value of f(x,y)


"f(1,1) = 6-9-3-7+5 = -8"




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