Answer to Question #99856 in Microeconomics for Tsige

Question #99856
Max(min) f(x,y,z) = x^2+y^2+z^2
Subject to
X+2y+z=30
2x-y-3z= 10
Check for local second order condition
1
Expert's answer
2019-12-03T10:13:14-0500

Solution:


"x=30-2y-z"


"2(30-2y-z)-y-3z=10"

"60-4y-2z-y-3z=10"

"50-5y-5z=0"

"10-y-z=0"

"y=10-z"

"x=30-2\\times (10-z)-z"

"x=30-20+2z-z"


"x=10+z"

"f(x;y;z)\\to f(z)"


"f(z)=(10+z)^2+(10-z)^2+z^2"


"f(z)=100+20z+z^2+100-20z+z^2+z^2"


"f(z)=3z^2+200"

"f'(z)=6z"

"6z=0; z=0"

With this value, a minimum of function is achieved.



"x=10; y=10"

"f(x;y;z)=10^2+10^2+0^2=200"

Answer: 200, minimum


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