Question

Find the least value of the expression $x^2+4y^2+3z^2-2x-12y-6z+14$.

Answer 1

Que. least value

$x^2+4y^2+{32}^2-2x-12y-62+14$

$\to =(x^2-2x+1)+4(y^2-2\times \frac{3}{2}\times 4+\frac{9}{4})+3(2^2-22+1)+14-1-9-3$

$=(x+1{)}^2+4(y-3/2{)}^2+3(2-1{)}^2+(2-1{)}^2+1$

since the first three term of this can not be

negative. $(\because (-a{)}^2=a^2)$

the smallest value of Question is 1

so the answer is 1