Problem

Compare the algebraic and contextual price domains of \(R(p)=p(40-p)\). Begin with the polynomial's all-real domain, solve the nonnegative price and demand constraints, state their intersection with endpoint status, and interpret the \(\text{two}~\text{included}~\text{boundaries}\).

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algebraic/unrestricted domain=(-∞,∞); contextual constraints=p≥0 only; intersection=p≥0; contextual domain=[0,∞) price units; endpoint status=0 included and no upper endpoint; retained graph=all points with nonnegative price

algebraic/unrestricted domain=[0,40]; contextual constraints=p≥0 and40−p≥0; intersection=0≤p≤40; contextual domain=[0,40] price units; endpoint status=both included; retained graph=entire parabola

algebraic/unrestricted domain=(-∞,∞); contextual constraints=p>0 and40−p>0; intersection=0<p<40; contextual domain=(0,40) price units; endpoint status=both excluded because revenue is zero; retained graph=open segment between the intercepts

algebraic/unrestricted domain=(-∞,∞); contextual constraints=p≥0 and40−p≥0; intersection=0≤p≤40; contextual domain=[0,40] price units; endpoint status=both included as zero-revenue boundaries

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Compare the algebraic and contextual price domains of \(R(p)=p(40-p)\). Begin with the polynomial's all-real domain, solve the nonnegative price and demand constraints, state their intersection with endpoint status, and interpret the \(\text{two}~\text{included}~\text{boundaries}\).

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