Is there a good polynomial approximation of $x \log(1/x)$ on $[0,1]$? The function $f(x)=x \log(1/x)$ is bounded on $[0,1]$ with $f(0)=f(1)=x$ as shown in the picture. Is there a general way to find a "good" polynomial approximation (or a polynomial of $x$ to fractional powers) of $f$ on the whole interval $[0,1]$?

 A: Just for fun. 
I have tried to interpolate $f_1(x)=x\log(1/x)$ in a number of ways. Basically, I was trying to find the best fit as a linear combination of functions.
For example, if you choose the following set of functions of $x$:
$$1, x, x^2, x^3, x^4, x^5, x^6$$ 
...Mathematica says that the best fit is:
$$f_2(x)=-9.50939 x^6+32.623 x^5-44.5309 x^4+31.3466 x^3-13.2117 x^2+3.27045 x+0.00929563$$
If you plot the error of approximation $\Delta =f_2(x)-f_1(x)$ you get the following graph:

Basically, even with the polynom of 6th degree you have an error of approximation of approximatelly 0.01. 
Not bad but I wanted something better. If you extend the set to $x^{10}$ (polynomial of 10th degree), the error goes down to approx 0.0015. I expected to see much smaller error so I turned to something else. 
I have tried with the following set:
$$x^{1/2}, x, x^{3/2}$$
The result turned out to be:
$$f_2(x)=-1.5206 x^{3/2}+0.877692 x+0.647164 \sqrt{x}-0.0115326$$
...and the error was:

So linear combination of these 3 functions has slightly bigger error of approximation (around 0.015) compared with a polynomial of the six degree.
I have then tried different sets of the form:
$$x^{\frac{n-1}n}, x, x^{\frac{n+1}n}$$
...and found out that the error of approximation goes down with $n$ increasing. For example, for $n=100$ the best fit is:
$$f_2(x)=49.599 x^{99/100}-50.4042 x^{101/100}+0.805204 x-5.38214\times 10 ^{-6}$$
...with the error of approximation no greater than approx. $5\times 10^{-6}$.

When you plot both functions, $f_1,f_2$, you have almost ideal overlapping:

