Ag.algebraic geometry – Purely inseparable morphism characterization

I have a question about Exercise 5.3.9 from Liu's "Algebraic Geometry" (page 208):

Enter image description here

Unfortunately, I have no idea how to (a) close the part

$ K (Y) to K (X) $ pure inseparable implies $ f $ purely inseparable.

By definition $ f: X to Y $ is called purely inseparable if $ f $ injective and if for everyone $ x ∈ X $, the extension of residual fields $ k (f (x)) → k (x) $
is a purely inseparable extension.

My thoughts:

We assume that $ K (X) / K (Y) $ is a finite pure inseparable extension.

Using the property of morphisms of annular spaces I get the following diagram

$$
require {AMScd}
begin {CD}
O_ {Y, f (x)} @> {} >> O_ {X, x} \
@VVV @VVV \
K (Y) = O_ {Y, eta_Y} @> {} >> K (X) = O_ {X, eta_X}
end {CD}
$$

By finiteness & pure separateness of field extension $ K (X) / K (Y) $ we obtain $ K (X) = K (Y)[a^{1/n}]$ for appropriate $ a in K (Y) $, The question is how this structure can be "translated" $ O_ {Y, f (x)} / m_ {f (x)} = k (f (x)) to k (x) = O_ {X, x} / m_x $?

On the other hand, there is another characterization of pure inseparable morphisms that applies to each area $ K $ the induced card $ X (K) to Y (K) $ is injective (see EGA 1 or Lius Ex. 3.8 from the previous page). But this approach does not help me either.

Does anyone know how I can attack the problem?

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