Question

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

Answer 1

`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