Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$ Greatest value of $f(x)= (x+1)^{1/3}-(x-1)^{1/3}$ on $(0,1)$
Please guide me to solve this problem. I have differentiated it with respect to $x$ and make equal to zero, but couldn't get any point.

By Jensen $$\left(\frac{\sqrt[3]{1+x}+\sqrt[3]{1-x}}{2}\right)^3\leq\frac{1+x+1-x}{2}=1,$$
which gives $\sqrt[3]{1+x}+\sqrt[3]{1-x}\leq2$.
The equality occurs for $x=0$.
Id est, the answer is $2$.
We used the following inequality.
For non-negatives $a$ and $b$ we have:
$$\left(\frac{a+b}{2}\right)^3\leq\frac{a^3+b^3}{2},$$
which is $(a+b)(a-b)^2\geq0$.
Also we can use P_M.
