Question

find the least value of the expression 4x^2 + 2y^2 - 4xy - 4y + 20

Answer 1

Answer:

x²+4y²+3z²-2x-12y-6z+14

=x²-2x+4y²-12y+3z²-6z+14

=x²-2x+4(y²-3y)+3(z²-2z)+ 14

=(x²-2x+1–1)+4{y²-2× (3/2)y+(3/2)²-(3/2)²} +3(z²-2z+1–1)+14

=(x²-2x+1)+4{y²-2 (3/2)y+(3/2)²}+3(z²-2x+1)-1–4×(3/2)²-3+14

=(x-1)²+4(y-3/2)²+3(z-1)²-1–4 ×9/4-3+14

=(x-1)²+4(y-3/2)²+3(z-1)+1

Hence when x=1, y=3/2 and z=1

Minimum value of the expression is = 1