Where $f(x_1,x_2,x_3)=2x_1^2+x_2^2+3x_3^2+2tx_1x_2+2x_1x_3$.
This is a problem in my Matrix Analysis homework. Below is my effort.
Let $x=(x_1,x_2,x_3)^T$, then we have $f=x^*Sx$, in which $S=\left(\begin{matrix}2&t&1\\t&1&0\\1&0&3\end{matrix}\right)$. $f$ is positive definite is equivalent to $S$ is positive definite which is equivalent to all the eigenvalues of $S$ is positive.
The characteristic polynomial of $S$ is: $\begin{align}|\lambda I-S|&=-\lambda^3+6\lambda^2+(3t^2-10)\lambda+(-3t^2+5)\\&=(-3+3\lambda)t^2+(-\lambda^3+6\lambda^2-10\lambda+5)\end{align}$.
Now the only problem left is that how do I find all the possible real values of $t$ that makes this polynomial of $\lambda$ only has positive roots?