Question

If $ab=25$, find the point where
$\Delta \left(x\right)=\left|\begin{array}{ccc}x& a& a\\ b& x& a\\ b& b& x\end{array}\right|$ has a local minimum.

Answer 1

$\Delta \left(x\right)=\left|\begin{array}{ccc}x& a& a\\ b& x& a\\ b& b& x\end{array}\right|andab=25\Delta \left(x\right)=x^3-3abx+a^{2}b+a{b}^{2}equatingfirstderivativetozerogives{\Delta }^{\prime }\left(x\right)=3x^2-3ab=0\Rightarrow x=\pm 5atx=5{\Delta }^{\prime \prime }\left(x\right)>0Hence\Delta \left(x\right)hasalocalminimumatx=5$