proving $t^6-t^5+t^4-t^3+t^2-t+0.4>0$ for all real $t$ proving $t^6-t^5+t^4-t^3+t^2-t+0.4>0$ for all real $t$
for $t\leq 1,$ left side expression is $>0$
for $t\geq 1,$ left side expression $t^5(t-1)+t^3(t-1)+t(t-1)+0.4$ is $>0$
i wan,t be able to prove for $0<t<1,$ could  some help me with this

Let $p(t) = t^6 - t^5 + t^4 - t^3 + t^2 - t +2/5$. Observe that
$$ p(t) = \begin{bmatrix} 1\\t\\t^2\\t^3\end{bmatrix}^\intercal \begin{bmatrix}2/5&-1/2&0&0\\-1/2&1&-1/2&0\\0&-1/2&1&-1/2\\0&0&-1/2&1\end{bmatrix}\begin{bmatrix} 1\\t\\t^2\\t^3\end{bmatrix}
$$
The matrix in the middle is positive definite, from which it follows immediately that $p(t) > 0$ for all $t$.
Edit: Positive definiteness can be determined by mechanically calculating the matrix's minors, which is easy.
