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I am trying to find the maximum of a fuction: $ f(x)=sqrtx^2+9-sqrtx^2-sqrt3 x+1 $.

I tried

 Maximize[Sqrt[x^2 + 9] - Sqrt[x^2 - Sqrt[3]*x + 1], x]

and I got

[Sqrt](9 +
Root[-3 + #1^2 &,
243 - 567 #1 #2 + 1557 #2^2 - 702 #1 #2^3 + 485 #2^4 -
71 #1 #2^5 + 35 #2^6 &, 2, 2]^2) - [Sqrt](1 -
Sqrt[3] Root[-3 + #1^2 &,
243 - 567 #1 #2 + 1557 #2^2 - 702 #1 #2^3 + 485 #2^4 -
71 #1 #2^5 + 35 #2^6 &, 2, 2] +
Root[-3 + #1^2 &,
243 - 567 #1 #2 + 1557 #2^2 - 702 #1 #2^3 + 485 #2^4 -
71 #1 #2^5 + 35 #2^6 &, 2, 2]^2), x ->
Root[-3 + #1^2 &,
243 - 567 #1 #2 + 1557 #2^2 - 702 #1 #2^3 + 485 #2^4 -
71 #1 #2^5 + 35 #2^6 &, 2, 2]

The exact answer is $sqrt7$.

How can I get the exact answer of the maximum of this function?

First set the function definition

Clear[f]
f[x_] := Sqrt[x^2 + 9] - Sqrt[x^2 - Sqrt[3] x + 1]

Method I

The conventional method, by finding the stationary point first:

Solve[f'[x] == 0, x][[1]]
f[x] /. %
(f''[x] /. %%) < 0

x -> (3 Sqrt[3])/5
Sqrt[7]
True

Method II

By using ToRadicals to transform Root:

Maximize[f[x], x] // ToRadicals

Sqrt[7], x -> (3 Sqrt[3])/5