For a discontinuous function ,maximun value is found by replacing every region for the values x ,y
So then for x≤2. and y≤1 and f(x,y) = x +y
theres a maximum value but no minimum ( because is minus infinite). The maximum value for this area is
x+y = 2+1= 3
Now for area y ≥-2x -3
rearrange the equation as y+2x ≥ -3
is the equation of a plane , the plane or area that goes over the straight line y= -2x-3.
So now replace y = -2x-3 in the function f(x,y) = x+y
then f(x,y)= x -2x-3 = -x-3
Now the third region is y≥ 2x -3
this is the area over a line with slope positive equal to 2
now replacing again for this area we find
f(x,y) = x+y = 2x-3 +x = 3x -3
because x≤2. So then replace x by 2 in f(x,y) to find the maximum value
f(x,y) in 2nd region is f(x,y) = -x-3= -2-3= -5
f(x,y) in 3rd region is f(x,y) = 3x -3= 3•2 -3= 3
Finally in conclusion, the máximum value is 3 , the minimum value is -5