Respuesta :
The gradient of this function is
[tex]\left(\dfrac{\partial z}{\partial x},\dfrac{\partial z}{\partial y}\right)=(4, 3)[/tex]
So, it is never zero, and the function has no absolute maximum/minimum points (after all, it's a plane...)
So, we evaluate the function along the borders of our domain, and look for the maximum value there. The domain is the trapezium with vertices
[tex]A=(2, 0),\ B=(6, 0),\ C=(6, 1),\ D=(2, 7)[/tex]
If we go along AB, y is constantly 0, and x increases from 2 to 6. So, the maximum value for [tex]4x+3y=4x[/tex] is 4*6=24
If we go along BC, x is constantly 6, and y increases from 0 to 1. So, the maximum value for [tex]4x+3y=24+3y[/tex] is 24+3*1=27
If we go along AD, x is constantly 2, and y increases from 0 to 7. So, the maximum value for [tex]4x+3y=8+3y[/tex] is 8+3*7=29
Finally, if we go along CD, y is -3/2x+10, and x ranges from 2 to 6. Since the line is descending, it has its maximum value at x=2, and we have point D again.
So, the maximum is 29, at (2,7).
