Đầu tiên ta có: 0 < z < 2\(\sqrt{5}\) ⇒ 20−z2 > 0, 3(9−2z) > 0, B−z > 0
\(5x^2+2xyz+4y^2+3z^2=60\)
\(\Leftrightarrow5\left(B-y-z\right)^2+2\left(B-y-z\right)yz+4y^2+3z^2=60\)
\(\Leftrightarrow\left(9-2z\right)y^2-2\left(B-z\right)\left(5-z\right)y+5\left(B-z\right)^2+3\left(z^2-20\right)=0\)
Đế pt theo nghiệm y có nghiệm thì
\(\Delta'=\left(B-z\right)^2\left(5-z\right)^2-\left(9-2z\right)\left(5\left(B-z\right)^2+3\left(z^2-20\right)\right)\ge0\)
\(\Leftrightarrow\left(z^2-20\right)\left(\left(B-z\right)^2-27+6z\right)\ge0\)
\(\Rightarrow\left(B-z\right)^2-27+6z\le0\)
\(\Rightarrow B\le z+\sqrt{27-6z}\le6\)
B đạt Max là 6 khi x = 1; y = 2; z = 3
