Quadratic Functions, Equations and Inequalities - Completing the Square and Vertex Form

A company manufactures and sells smartphones. The profit \(P\) (in million dollars) from producing and selling \(x\)(in ten thousands) smartphones per month is given by the quadratic equation:

\(\mathrm{P}\left( x \right)=-2{{x}^{2}}+40x-150\)

1. What is the initial investment for the company before producing any smartphones?
2. Express \(P\) in form of \(P=a{{\left( x-h \right)}^{2}}+k\) where \(a\), \(h\) and \(k\) are constants to be determined.
3. Determine the number of smartphones the company needs to produce and sell each month to maximise profit.
   The company is experimenting with a new model of smartphones. The profit \(Q\) (in million dollars) from producing and selling \(x\)(in ten thousands) smartphones per month is given by the quadratic equation:
   \(\mathrm{Q}\left( x \right)=-2{{x}^{2}}+40x-220\)
4. Explain why it is not feasible for the company to produce the new model of smartphones.