I am supposed to study the function $$f(x)=x\cdot \ln(x+2)$$ To find the maxima and minima, I am supposed to set its first derivative $$f'(x)=\ln(x+2) + \frac{x}{x+2} \ge 0$$
However, I am unsure of how to solve this. Is there an easy (that I could have reasonably been taught given that I am in my last year of high school), that I am overlooking, to solve this, or are there any alternative methods/shortcuts?
Thanks in advance
In order to solve $f'(x)=0$ you have to use Lambert W function or numerical methods like Newtons method. I am solving via Lambert W function,
$$\ln(x+2) + \frac{x}{x+2} = 0{ \implies\ln y +\frac{y-2}y=0\quad(\text{Where, $y=x+2$})\\ \implies y(\ln y+1)=2\\ \implies y\ln(ey)=2\\ \implies \ln(ey)=\frac{2}y\\ \implies 2e=\frac{2}y e^{2/y}\\ \implies W(2e)=\frac{2}y\\ \implies y=\frac{2}{W(2e)}\\ \implies x=\frac{2}{W(2e)}-2\approx-0.5453}$$
For high school student I recommend to solve the equation by graph, plot $f(x)=\ln(x+2)$ and $g(x)=-\frac{x}{x+2}$ and find the intersection point.