Question

Determine the intervals of monotonicity of the function $f\left(x\right)=\left|\begin{array}{ccc}x+a^2& ab& ac\\ ab& x+b^2& bc\\ ac& ba& x+c^2\end{array}\right|$

• A. $xϵ(-\infty ,l)\cup (0,\infty ),l=-\frac{5}{3}\sum a^2$
• B. $xϵ(-\infty ,l)\cup (0,\infty ),l=-\frac{4}{3}\sum a^2$
• C. $xϵ(-\infty ,l)\cup (0,\infty ),l=-\frac{1}{3}\sum a^2$
• D. $xϵ(-\infty ,l)\cup (0,\infty ),l=-\frac{2}{3}\sum a^2$

Answer 1

The correct option is D $xϵ(-\infty ,l)\cup (0,\infty ),l=-\frac{2}{3}\sum a^2$

By rule for differentiation of determinants
$f^{\prime }\left(x\right)=\left|\begin{array}{ccc}1& 0& 0\\ ab& x+b^2& bc\\ ac& b& x+c^2\end{array}\right|+\left|\begin{array}{ccc}x+a^2& ab& ac\\ 0& 1& 0\\ ac& bc& x+c^2\end{array}\right|+\left|\begin{array}{ccc}x+a^2& ab& ac\\ ab& x+b^2& bc\\ 0& 0& 1\end{array}\right|$

or $f^{\prime }\left(x\right)=[(x+b^2)(x+c^2)-b^2{c}^2]+\left[(x+a^2)(x+c^2-a^2{c}^2)\right]+\left[(x^2+a^2)(x^2+b^2-a^2{b}^2)\right]$

or $f^{\prime }\left(x\right)=3x^2+2x(a^2+b^2+c^2)$ or
$f\left(x\right)=3x(x+\frac{2}{3}\sum a^2)=3[x-(-\frac{2}{3}\sum a^2)][x-0]$
$=3[x-l][x-0],l=-\frac{2}{3}\sum a^2$

for $F\left(x\right)$ is decreasing. $f^{\prime }\left(x\right)<0\Rightarrow xϵ(l,0)$
and for $f\left(x\right)$ is increasing.

$f^{\prime }\left(x\right)=+ive$ when either $x<lorx>0$
$\therefore xϵ(-\infty ,l)orxϵ(0,\infty )$
$\therefore f\left(x\right)$ is increasing
when $xϵ(-\infty ,l)\cup (0,\infty ),$