Question

If $\Delta >0$, then

• A. $f\left(1\right)>0$
• B. $f\left(1\right)<0$
• C. $f\left(1\right)=0$
• D. cannot say anything about f(1)

Answer 1

The correct option is D cannot say anything about f(1)

$\Delta =\left|\begin{array}{ccc}3& 1+S_1& 1+S_2\\ 1+S_1& 1+S_2& 1+S_3\\ 1+S_2& 1+S_3& 1+S_4\end{array}\right|=\left|\begin{array}{ccc}3& 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2\\ 1+\alpha +\beta & 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3\\ 1+{\alpha }^2+{\beta }^2& 1+{\alpha }^3+{\beta }^3& 1+{\alpha }^4+{\beta }^4\end{array}\right|=\left|\begin{array}{ccc}1& 1& 1\\ 1& \alpha & \beta \\ 1& {\alpha }^2& {\beta }^2\end{array}\right|\left|\begin{array}{ccc}1& 1& 1\\ 1& \alpha & \beta \\ 1& {\alpha }^2& {\beta }^2\end{array}\right|={\left|\begin{array}{ccc}1& 1& 1\\ 1& \alpha & \beta \\ 1& {\alpha }^2& {\beta }^2\end{array}\right|}^2={(1-\alpha )}^2{(1-\beta )}^2{(\alpha -\beta )}^2$
Can't say anything about f(1)