7.3 Optimizing Functions of Several Variables Page 457 Critical Point (?, ?)is a critical point of 𝑓(?, ?)if 𝑓?(?, ?) = 0and 𝑓?(?, ?) = 0Relative maximumand minimumvalues can occur only at critical points.
Ex) Find all critical points: 𝑓(?, ?) = 2?2+ ?2+ 2?? + 4? + 2? + 5Second-Derivative Test for Functions 𝑓(?, ?): The D-Test If (?, ?)is a critical point of the function 𝑓and if ?is defined by ? = 𝑓??(?, ?) ∙ 𝑓??(?, ?) − [𝑓??(?, ?)]2Then 𝑓at the point (?, ?)has a : i. relativemaximumif ? > 0and 𝑓??(?, ?) < 0ii. relative minimumif ? > 0and 𝑓??(?, ?) > 0iii. Saddle pointif ? < 0
#22 (Page 466) BUSINESS: Maximum Profit A company manufactures two products. The price function of product A is ? = 16 − ?(for 0 ≤ ? ≤ 16), and product B is ? = 19 −2?(for 0 ≤ ? ≤ 38), both in thousands of dollars, where ?and ?are the amounts of product A and B, respectively. If the cost function is ?(?, ?) = 10? + 12? − ?? + 61Thousand dollars, find the quantities and the prices of the two products that maximize profit. Also find the maximum profit.