Question

If a,b > 0 and
$\Delta$ = $\left|\begin{array}{ccc}x& a& a\\ b& x& a\\ b& b& x\end{array}\right|$ then
$\Delta$ has a local minimum at $x$ equal

• A. $\sqrt{ab}$
• B. $-\sqrt{ab}$
• C. $\sqrt{\frac{a}{b}}$
• D. $-\sqrt{\frac{a}{b}}$

Answer 1

The correct option is A $\sqrt{ab}$

$\Delta =\left|\begin{array}{ccc}x& a& a\\ b& x& a\\ b& b& x\end{array}\right|$
so, $\Delta =x(x^2-ab)-a(bx-ab)+a(b^2-bx)$
$\Delta =x^3-3abx+a^{2}b+a{b}^2$
$\Delta$ is a cubic in $x$
so, for local minimum of $\Delta$,
$\frac{d\Delta }{dx}=3x^2-3ab=0$
$x=\sqrt{ab}$