Fixed-Point Theorem Question

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I am reading solution to this problem and I have questions.

In the fix point function in the problem is:

$g(x) = (3x^2 + 3)^\frac{1}{4}$ and where x $\in$ [1,2]

And g prime is in the image I linked to.

enter image description here

My question is why is $k = 6^{-.5}$

That is, where does it come from?