Among all pairs of numbers X Y such that X2Y4 what is the ma

Among all pairs of numbers X, Y such that X+2Y=4 what is the maximum or minimum value of X²+XY.

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Z = X^2 +XY

Now we have X +2Y = 4

X = 4- 2Y

So, Z = (4-2Y)^2 +(4-2Y)Y

= 16 +4y^2 - 16y +4y -2y^2

= 2y^2 -12y +16

Its a qadratice equation of the form : ax^2 +bx +c

Its max./min. would occur at vertex: x= -b/2a

So, in our case Y = -b/2a = -(-12)/(2*2) = 3

X = 4-2Y = 4 -2*3 = 4-6 = -2

So, (X , Y) = ( -2, 3 )