Critical point of a function? Can anyone show me how to simplify get the critical point of this function.
$f(x)=x^2\sqrt[3]{2+x}$
I did the product rule and got
$$2x\sqrt[3]{2+x}+\frac{x^2}{3\sqrt[3]{(2+x)^2}}$$
but I am having touble simplifying such this how would I simplify it can anyone show me how this would be done. 
 A: You need only find the points at which $f'(x) = 0$, and where $f'(x)$ is undefined.
To simplify, find a common denominator and add terms, then set equal to zero: The common denominator is $3\sqrt[\large 3]{(2 + x)^2}$. Do you recall how to bring all terms over this common denominator? 
We have $$2x\sqrt[\large3]{2+x}+\frac{x^2}{3\sqrt[\large3]{(2+x)^2}}$$
And want 
$$
\begin{align} f'(x) 
& = \dfrac{2x \sqrt[\large 3]{2+x}\times 3\sqrt[\large 3]{(2+x)^2} + x^2}{3\sqrt[\large 3]{(2+x)^2}}  \\ \\
& = \frac{6x \sqrt[\large 3]{(2+x)(2+x)^2} + x^2}{3 \sqrt[\large 3]{(2+x)^2}} \\ \\
& = \frac{6x \sqrt[\large 3]{(2+x)^3} + x^2}{3 \sqrt[\large 3]{(2+x)^2}} \\ \\
& = \dfrac{6x(2 + x) + x^2}{3\sqrt[\large 3]{(2+x)^2}} \\ \\
& = \dfrac{12x + 6x^2 + x^2}{3 \sqrt[\large 3]{(2+x)^2}} \\ \\
& = \dfrac{x(12 + 7x)}{3\sqrt[\large 3]{(2 + x)^2}} = 0 \\ \\
\end{align}
$$
$f'(x)$ is undefined when $x = -2$.
$f'(x) = 0$ when the numerator equals zero: one point at which this occurs is when $x = 0$.
$f'(x) = 0$ when $(12 + 7x) =0 \iff 7x = -12 \iff x = -\large\frac{12}{7}$
Three critical points in all. The blue line below is your function of interest. Note the sharp corner at $x = -2$. It happens to be a local minimum. Also note the local maximum at $x = -\large\frac{12}{7}$, and the  minimum at $x = 0$. Graphs are really helpful to confirm the work you're doing, and better understand the behavior of the function.


ASIDE: Personally I think using fractional exponents to express roots makes this sort of problem a bit clearer, in terms of algebraic manipulation, particularly when we're talking about roots other than the square root, and especially when they appear in fractions.
A: On simplification $$f'(x)=\frac{6x(2+x)^{3/3}+x^2}{3(2+x)^{2/3}}=\frac{12x+7x^2}{3(2+x)^{2/3}}$$
Now, critical poits are where $f'(x)=0$ which gives $x=0,x=-12/7$
and those points where $f'(x)$ is not defined like $x=-2$
