Question

If $a$, $b$, $c$ are sides of a scalene triangle, then the value of $\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|$ is

• A. non-negative
• B. negative
• C. positive
• D. non-positive

Answer 1

The correct option is D non-positive

$\left|\begin{array}{ccc}a& b& c\\ b& c& a\\ c& a& b\end{array}\right|=a(bc-a^2)-b(b^2-ac)+c(ab-c^2)$

$=abc-a^3-b^3+abc+abc-c^3$

$=3abc-a^3-b^3-c^3\le 0$

we know that,

$3abc\le (a^3+b^3+c^3)$ for any non-negative $a,b,c$

Hence it is non-positive

Option D is correct