Find positive weights (w) in a utility function of the form (uleft(c_{1}, c_{2} ight)=) (c_{1}+w cdot c_{2}) to create at least four different strict rankings

Find positive weights \(w\) in a utility function of the form \(u\left(c_{1}, c_{2}\right)=\) \(c_{1}+w \cdot c_{2}\) to create at least four different strict rankings among the candidates \(x, y, z\) in (5.10). [Hint: Experiment with some values of \(w\) and vary them. What are the values of \(u\left(c_{1}, c_{2}\right)\) for the three candidates as a function of \(w\) ?]