## 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 \(\mathbf u\)?