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Sets and absolute minimum

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Let S = {(x,y,z):2x-y+3z=6},p=(1,0,-1) belonging to R3, and let f:S->R be defined by f(q)=d(q,p),

ie, f(x,y,z)=sqrt((x-1)^2+y^2+(z+1)^2)

Question: Is S compact? Please verify if q belongs to S and q does NOT belong to [-5,5]^3, then f(q) >= 4.

ALSO prove that f attains its absolute minimum value at some point q0 belonging to S, and that q0 must be inside [-5,5]^3. (for this question I know you can use f(0,0,2) = sqrt(10) < 4)

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Solution Summary

This shows how to identify if S is compact, if a given term belongs to S, and also proves that a function attains absolute minimum value at a point in set S

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