Question

In which of the following interval $f\left(x\right)$ is strictly increasing?

• A. $(-\infty ,\infty )$
• B. $(-\infty ,0)$
• C. $(0,\infty )$
• D. none of these

Answer 1

The correct option is C $(0,\infty )$

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

$\frac{1}{abc}\left|\begin{array}{ccc}a(a^2+x)& a^{2}b& a^{2}c\\ a{b}^2& b(b^2+x)& b^{2}c\\ a{c}^2& b{c}^2& c(c^2+x)\end{array}\right|$

$\left|\begin{array}{ccc}(a^2+x)& a^2& a^2\\ b^2& (b^2+x)& b^2\\ c^2& c^2& (c^2+x)\end{array}\right|$

$R_1\to R_1+R_2+R_3$

$\left|\begin{array}{ccc}(a^2+b^2+c^2+x)& (a^2+b^2+c^2+x)& (a^2+b^2+c^2+x)\\ b^2& (b^2+x)& b^2\\ c^2& c^2& (c^2+x)\end{array}\right|$

$(a^2+b^2+c^2+x)\left|\begin{array}{ccc}1& 1& 1\\ b^2& (b^2+x)& b^2\\ c^2& c^2& (c^2+x)\end{array}\right|$

$C_1\to C_1-C_3$

$(a^2+b^2+c^2+x)\left|\begin{array}{ccc}0& 1& 1\\ 0& (b^2+x)& b^2\\ -x& c^2& (c^2+x)\end{array}\right|$

$\Rightarrow f\left(x\right)=x^2(a^2+b^2+c^2+x)$

Now, $f^{\prime }\left(x\right)=x^2+2x(a^2+b^2+c^2+x)$

$\Rightarrow f^{\prime }\left(x\right)=3x^2+2x(a^2+b^2+c^2)$

$\Rightarrow f^{\prime }\left(x\right)=x(3x+2(a^2+b^2+c^2\left)\right)$

For strictly increasing , $f^{\prime }\left(x\right)>0$

$\Rightarrow x(3x+2(a^2+b^2+c^2\left)\right)>0$

$\Rightarrow x>0$

Therefore, $f\left(x\right)$ is increasing in $(0,\infty )$

Hence, option 'C' is correct.