I'm trying to determine what (real) values of $b$ the quadratic form $q = bx_1^2 + 2bx_2^2 + (9b + 2)x_3^2 − 2bx_1x_2 − 6bx_1x_3 + 4bx_2x_3$.
I know (using leading principal of minors) that the quadratic form is positive definite for $b \in (0,2)$ and positive semi-definite for $b \in [0,2]$ and that it cannot be negative definite/semi-definite for any real values of $b$. Thus this would indicate that it's indefinite for $b \in \mathbb R \text{\\} [0,2]$, however I am told that the answer is $b \in \mathbb R$. I can't see why this is correct as the quadratic form will be positive definite for $b \in (0,2) \subset \mathbb R$.
The quadratic form corresponds to the matrix $$ \begin{bmatrix} b&-b&-3b\\ -b&2b&2b\\ -3b&2b&9b+2 \end{bmatrix} $$ Computing the eigenvalues of this matrix for $b=1$ gives $$ \lambda=\begin{bmatrix} 12.2992\\ 1.65159\\ 0.049229 \end{bmatrix} $$ i.e. the quadratic form is positive definite, whereas for $b=3$ we get $$ \lambda=\begin{bmatrix} 33.4129\\ 4.64512\\ -0.0579871 \end{bmatrix} $$ i.e. the quadratic form is indefinite.
I think your answer is correct.