Perhaps this

n = 10; v0 = 20;
y = Sum[(v[i] - v0/Xs*(Xs - x[i])^3)^2, {i, 0, n}];
y1 = Expand[D[y, Xs] Xs^3];(*Remove Xs from denominators*)
Simplify[Reduce[y1 == 0, Xs]]

All of the details of determining the sixth power polynomial in Xs has been done for you. With your values for x[i] and v[j] perhaps you can determine which roots are Real.

Or if the Root notation is confusing then use your values of x[i] and v[j] to determine the real roots of

440 Xs^6 -
Xs^5 Sum[180 x[i], {i, 0, 10}] -
Xs^4 Sum[2 v[i] - 300 x[i]^2, {i, 0, 10}] +
Xs^3 Sum[3 x[i] v[i] - 200 x[i]^3, {i, 0, 10}] -
Xs Sum[x[i]^3 v[i] - 60 x[i]^5, {i, 0, 10}] -
Sum[20 x[i]^6, {i, 0, 10}]

which is just a manual reformatting of the result from the code above.

Plot might even help understanding how many real roots there are or Reduce might do that if you define all your x[i] and v[j] values first.