Sapphire begins 2018 with 980 comma 000 12-ounce bottles in inventory. The vice president of operations requests that 12-ounce bottles ending inventory on December 31, 2018, be no less than 660 comma 000 bottles. Based on sales projections as budgeted previously, what is the minimum number of 12-ounce bottles Sapphire must produce during 2018

David consumes two things: gasoline (g) and bread (b). david's utility function is u(g, b) = 10g^0.25 b^0.75. use the lagrange technique to solve for david's optimal choices of gasoline and bread as a function of the price of gasoline, p_g, the price of bread, p_b, and his income m. with recent decrease in the price of gasoline (maybe due to external shock such as shale gas production) does david increase his consumption of gasoline? for david, how does partial differential g/partial differential p_g depend on his income m? that is, how does david's change in gasoline consumption due to an increase in the price of gasoline depend on his income level? to answer these questions, find the cross-partial derivative, |partial differential^2 g/partial differential m partial differential p_g.