21. [-/2 Points] DETAILS MY NOTES SCALCET9 2.7 .052 . The quantity (in pounds) of a gourmet ground coffee that is sold by a coffee company at a price of \( p \) dollars per pound is \( Q=f(p) \). (a) What is the meaning of the derivative \( f^{\prime}(4) \) ? What are its units? The price of the coffee as a function of the supply. The units are pounds. The rate of change of the quantity of coffee sold with respect to the price per pound when the price is \( \$ 4 \) per pound. The units are pounds/(dollars/pound). The rate of change of the price per pound with respect to the quantity of coffee sold when the price is \( \$ 4 \) per pound. The units are dollars/pound. The rate of change of the quantity of coffee sold with respect to the price per pound when the price is \( \$ 4 \) per pound. The units are dollars/pound. The rate of change of the price per pound with respect to the quantity of coffee sold when the price is \( \$ 4 \) per pound. The units are pounds/(dollars/pound). (b) Is \( f^{\prime}(4) \) positive or negative? Explain. positive negative
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The derivative \( f^{\prime}(4) \) represents how the quantity of gourmet coffee sold changes as the price per pound of the coffee approaches \$4. If \( f^{\prime}(4) \) is positive, it means that as the price increases, the quantity sold also increases, suggesting a potential rise in demand. The units for this derivative are pounds per dollar, indicating how many pounds of coffee are sold for each additional dollar increase in price. When examining whether \( f^{\prime}(4) \) is positive or negative, consider typical market behavior: if the price of gourmet coffee rises, it's common for the demand to wane; thus, \( f^{\prime}(4) \) is likely negative. This negative derivative implies that higher prices result in lower quantities sold, aligning with the law of demand where price and quantity sold are inversely related in most cases.
