I have a function defined as

```
f(k_):= Maximize({(x^2/(1+x^2))((1-x^(2k))/(1+x^(2k+1)))^2, 0 <= x <= 1}, x)
```

I would like to find an analytic form of $f$ as a function of $k$.

Differentiating the argument of the Maximize in $f$ with respect to $x$, I obtain the condition:

```
1+x^(3+4k) - x^(2k)(1+x)(x^2(1+2k)- x + (1+2k)) == 0
```

Is there a way I can write a replacement rule in Mathematica to substitute the above condition on $x$, and get $f$ as a function of $k$ alone?