Giải thích các bước giải:
Ta có:
\(\begin{array}{l}
c,\\
8{x^2} + {y^2} + 11{z^2} + 4xy - 12xz - 5yz\\
= \left( {4{x^2} + 4xy + {y^2}} \right) - \left( {10xz + 5yz} \right) + 11{z^2} + 4{x^2} - 2xz\\
= {\left( {2x + y} \right)^2} - 5z.\left( {2x + y} \right) + \frac{{25}}{4}{z^2} + \frac{{19}}{4}{z^2} + 4{x^2} - 2xz\\
= \left[ {{{\left( {2x + y} \right)}^2} - 2.\left( {2x + y} \right).\frac{{5z}}{2} + {{\left( {\frac{{5z}}{2}} \right)}^2}} \right] + \left( {{x^2} - 2xz + {z^2}} \right) + 3{x^2} + \frac{{15}}{4}{z^2}\\
= {\left[ {\left( {2x + y} \right) - \frac{{5z}}{2}} \right]^2} + {\left( {x - z} \right)^2} + 3{x^2} + \frac{{15}}{4}{z^2} \ge 0,\,\,\,\forall x,y,z\\
d,\\
5{x^2} + 5{y^2} + 5{z^2} + 6xy - 8xz - 8yz\\
= \left( {3{x^2} + 6xy + 3{y^2}} \right) - \left( {8xz + 8yz} \right) + 2{x^2} + 2{y^2} + 5{z^2}\\
= 3.\left( {{x^2} + 2xy + {y^2}} \right) - 8z.\left( {x + y} \right) + 5{z^2} + 2{x^2} + 2{y^2}\\
= 3.{\left( {x + y} \right)^2} - 8z.\left( {x + y} \right) + 5{z^2} + 2{x^2} + 2{y^2}\\
= 3.\left[ {{{\left( {x + y} \right)}^2} - \frac{8}{3}z.\left( {x + y} \right) + \frac{{16}}{9}{z^2}} \right] + \frac{{29}}{9}{z^2} + 2{x^2} + 2{y^2}\\
= 3.\left[ {{{\left( {x + y} \right)}^2} - 2.\left( {x + y} \right).\frac{{4z}}{3} + {{\left( {\frac{{4z}}{3}} \right)}^2}} \right] + \frac{{29}}{9}{z^2} + 2{x^2} + 2{y^2}\\
= 3.{\left( {x + y - \frac{{4z}}{3}} \right)^2} + \frac{{29}}{9}{z^2} + 2{x^2} + 2{y^2} \ge 0,\,\,\forall x,y,z
\end{array}\)
