How do you maximize 3x+4y-yz, subject to x+y<4, z>3?

1 Answer
Cesareo R.
May 22, 2016

#infty#

Explanation:

#f(x,y,z)=3x+4y-y z# has not stationary points because there are no points obeying the condition

#grad f(x,y,z) = vec 0#

So their extrema could be located at the viable region frontiers. Taking a restriction frontier, for instance #g_2(x,y,z)=z-3=0# and substituting in #f(x,y,z)# we get

#f(x,y,z)_{g_2} = f_{g_2}(x,y)=3 x + y#

calculating #grad f_{g_2}(x,y) = {3,1}#
so also no stationary points over #z=3#

The reduced problem reads now
Maximize #f_{g_2}(x,y)=3 x + y# with the border restriction
#x+y=4#. Applying the same idea as before, substituting the border relation in the objective function, we attain
#(f_{g_2})_{g_1} = 3x+(4-x)=2x+4# we see that the value range for #(f_{g_2})_{g_1} #is unlimited

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