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I tried the following:

f[x_] := x^n/(k + x^(2 n));
Solve[f'[x] == 0, x]

But I got the following response:

Solve::nsmet: This system cannot be solved with the methods available to Solve. >>

Any suggestions?

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f[x_] := x^n/(k + x^(2 n));
Solve[Simplify[f'[x]] == 0, x]

(* ==> {{x -> k^(1/(2 n))}} *)
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Using Reduce and limiting the domain (Reals) often produces more information:

ClearAll[f,x,n,k]

f[x_] := x^n/(k + x^(2 n))

sol = Reduce[f'[x] == 0, x, Reals];

The head of sol is Or; formatted to be more readable:

List @@ sol // Column

Mathematica graphics

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