## 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 $$\mathbf u$$?