A minimum that depends on a parameter

7 written marks

For real \(k\), let \(f_k(x)=2x^2-4(k+1)x+2k^2+3\).

1. Write \(f_k(x)\) in completed-square form.

   [3]

2. Find all \(k\) such that \(f_k(x)\gt 0\) for every real \(x\).

   [2]

3. At the boundary value \(\displaystyle k=\frac14\), solve \(f_k(x)=0\) and describe the root.

   [2]

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