Find the critical numbers of the function. f(x) = x² log,(x? + 2)

$$\begin{gathered} f\left(x\right)=x^2{\log }_7\left(x^2+2\right) \\ \Rightarrow f\left(x\right)=\frac{x^2\log \left(x^2+2\right)}{\ln 7} \left[\because {\log }_{a}b=\frac{\log b}{\ln a}\right] \end{gathered}$$

Critical numbers are the values of x for which first derivative is zero.

So first we find the derivative of given function.

$\begin{array}{rcl}{f}^{/}\left(x\right)& =& \frac{1}{\ln 7}\left[\frac{d\left(x^2\right)}{dx}·\log \left(x^2+2\right)+\frac{d\left(\log \left(x^2+2\right)\right)}{dx}·x^2\right]\\ & =& \frac{1}{\ln 7}\left[2x·\log \left(x^2+2\right)+\frac{1}{x^2+2}·\frac{d\left(x^2+2\right)}{dx}·x^2\right]\\ & =& \frac{1}{\ln 7}\left[2x·\log \left(x^2+2\right)+\frac{1}{x^2+2}·2x·x^2\right]\\ & =& \frac{2x}{\ln 7}\left[\log \left(x^2+2\right)+\frac{x^2}{x^2+2}\right]\\ & =& \frac{2x}{\ln 7}\left[\frac{\left(x^2+2\right)\log \left(x^2+2\right)+x^2}{x^2+2}\right]\end{array}$