How can I get the exact maximum of a function with a symbolic iterator index?

Question

I have a functional sequence whose general term is $a_n=x^n-x^{2\,n}$ where $x\in [0,1]$ and $n$ is a symbolic iterator index, which means $n$ is an arbitrary positive integer. When $n$ runs from 1 to an arbitrary large positive integer, it gives the sequence.

I want to get the maximum of the general term $a_n$ with respect to the variable $x$ where $x\in [0,1]$ but with $n$ being kept in place, which means the solution should be values containing numbers,and/or $n$ but no $x$, illustrating the maximum of $a_n$ for every possible $n$.

The algorithm is simple, for every possible $n$, $n$ can be regarded as a positive constant in the term, then we can solve for the null point of the derivatives of $a_n=x^n-x^{2\,n}$ with respect to $x$:

Solve[D[x^n - x^(2 n), x] == 0, x]

{{x -> 2^(-1/n)}}

And the maximum:

After examining the sign of the derivative, it can be determined {{x -> 2^(-1/n)}} is the point where the general term attains its maximum.

Simplify[(x^n - x^(2 n)) /. x -> 2^(-1/n), n >= 1]

1/4

Although I can calculate step by step like above, what I want now is a function which can automate this process. Preferably, the built-in function Maximize[] since I have noticed:

Maximize[{x^\[Pi] - x^(2 \[Pi]), 0 <= x <= 1}, x]

{1/4, {x -> 2^(-1/[Pi])}}

where $\pi $ is a built-in constant greater than 1 and similar to $n$ in my sequence. However, when I try:

Maximize[{x^n - x^(2 n), 0 <= x <= 1}, x]

Maximize::infeas:"There are no values of {x} for which the constraints 0<=x<=1 are
satisfied and the objective function x^n-x^(2\n) is real-valued."

I guess it is because MMA does not know $n$ can be regarded as a constant for a specific step in my sequence, so I made those definitions below, intending to make $n$ just look like $\pi$ for MMA:

SetAttributes[n, Constant];
NumericQ[n] = True;
$Assumptions = n \[Element] Integers && n >= 1;

However, when I run the same code, it still gives:

Maximize::infeas:"There are no values of {x} for which the constraints 0<=x<=1 are
satisfied and the objective function x^n-x^(2\n) is real-valued."

My questions are:

1.Why Maximize[{x^\[Pi] - x^(2 \[Pi]), 0 <= x <= 1}, x] works but Maximize[{x^n - x^(2 n), 0 <= x <= 1}, x] fails even after I made those definitions?

2.How can I make MMA recognize in a specific scope that $n$ can be regarded as a positive integer constant and thus give me the correct answer {1/4, {x -> 2^(-1/n)}}when I run Maximize[{x^n - x^(2 n), 0 <= x <= 1}, x]?

3.Is there any other functions that can be somehow employed to automate the process and get the correct answer?

Thanks in advance.

Answer 1

Edit

Solution with Maximize.

Maximize works best with linear and polynomial expressions.( See help function) Therefore using variable transformations.

an = x^n - x^(2 n) // Factor

(*   -x^n (-1 + x^n)   *)

max = Maximize[{an /. x^n -> xn, 0 <= xn <= 1}, xn]

(*   {1/4, {xn -> 1/2}}   *)

Solve[(xn /. max[[2]]) == x^n && 0 <= x <= 1 && n > 0, x]

(*   {{x -> ConditionalExpression[2^(-1/n), n > 0]}}   *)

A solution with Reduce

(red = List @@ 
    Reduce[{an == x^n - x^(2 n), 0 <= x <= 1, n > 0, 
      Element[n, Integers]}, x, Reals, 
     Backsubstitution -> True]) // TableForm

Maximum for an is 1/4 at x == 2^(-1/n).

Didn't get Maximize to extract max of an from red.

Example from comment.

Since x and y are form 0 to Pi/2, eliminating Cos and converting Sin[x] == sx, Sin[y] == sy.

anequ = an==n(Sin[x] + Cos[y] + Cos[x - y])//TrigExpand

(*   (0 <= x <= \[Pi]/2, 0 <= y <= \[Pi]/2)    *)

eli = Eliminate[{anequ, Sin[x]^2 + Cos[x]^2 == 1, 
   Sin[y]^2 + Cos[y]^2 == 1, Sin[x] == sx, Sin[y] == sy}, {Cos[x], 
   Cos[y]}]

sol1 = First@
   Solve[{Sin[x] == sx, Sin[y] == sy, 0 <= x <= Pi/2, 
     0 <= y <= Pi/2}, {x, y}] // 
  Simplify[#, {0 <= x <= Pi/2, 0 <= y <= Pi/2}] &

eli2 = List @@ (eli /. sol1 // Simplify)

(*   {an^4 + 4 n^4 sx^2 (1 + sy)^2 (-1 + sx^2 + sy^2) + 
   4 an^2 n^2 (1 + sy) (-1 + sy + sx^2 (2 + sy)) == 
  4 an n sx (1 + sy) (an^2 + 2 n^2 (1 + sy) (-1 + sx^2 + sy)), 
 0 <= sy < 1 && 0 <= sx < 1}   *)

max = Maximize[{an, eli2 && n > 0}, {an, sx, sy}] // 
  Simplify[#, Assumptions -> Element[n, Integers] && n > 0] &

(*   {(3 Sqrt[3] n)/2, {an -> (3 Sqrt[3] n)/2, sx -> Sqrt[3]/2, sy -> 1/2}}   *)

{x -> ArcSin[sx], y -> ArcSin[sy]} /. max[[2]]

(*   {x -> Pi/3, y -> Pi/6}   *)

Answer 2

$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global`*"]

tab1 = Table[{n, MaxValue[{x^n - x^(2 n), 0 <= x <= 1}, x]}, 
  {n, 1, 10}]

(* {{1, 1/4}, {2, 1/4}, {3, 1/4}, {4, 1/4}, {5, 1/4}, {6, 1/4}, 
  {7, 1/4}, {8, 1/4}, {9, 1/4}, {10, 1/4}} *)

max[n_] = 1/4;

tab2 = Table[{n, ArgMax[{x^n - x^(2 n), 0 <= x <= 1}, x]}, 
  {n, 1, 10}]

(* {{1, 1/2}, {2, 1/Sqrt[2]}, {3, 1/2^(1/3)}, {4, 1/2^(1/4)}, 
  {5, 1/2^(1/5)}, {6, 1/2^(1/6)}, {7, 1/2^(1/7)}, {8, 1/2^(1/8)}, 
  {9, 1/2^(1/9)}, {10, 1/2^(1/10)}} *)

argMax[n_] = 2^FindSequenceFunction[Log2[tab2[[All, 2]]], n]

(* 2^(-1/n) *)