Positive definite form

Task number: 2739

Decide whether \(g(\mathbf u)>0\) holds for all nontrivial \(\mathbf u=(x_1, x_2, x_3)\in \mathbb R^3\).

  • Variant 1

    \(g(\mathbf u)=x_1^2+2x_1x_2+2x_1x_3+2x_2^2+5x_3^2\)

  • Variant 2

    \(g(\mathbf u)=x_1^2+2x_1x_2+2x_1x_3+2x_2^2+5x_3^2+2ax_2x_3\), solve w.r.t. the parameter \(a\).

    Does for all other choices of \(a\) hold that \(g(\mathbf u)\le 0\) for all \(\mathbf u\)?

Difficulty level: Easy task (using definitions and simple reasoning)
Solution require uncommon idea
Cs translation
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