Question

What is the smallest possible value of \( x^2 - 3x + 2y^2 + 4y + 2 \)

Answer 1

Spivak suggests completing the square again

Answer 2

(x^2 - 3x + 9/4) + 2y^2 + 4y + 2 - 9/4
= (x - 3/2)^2 + 2(y^2 + 2y + 1) - 9/4
= (x - 3/2)^2 + 2(y+1)^2 - 9/4

seems to be -9/4