130Lec7-3 Optimizing Functions - 7.3 Optimizing Functions of Several Variables Page 457 Critical Point is a critical point of if = 0 and = 0

130Lec7-3 Optimizing Functions - 7.3 Optimizing...

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7.3 Optimizing Functions of Several Variables Page 457 Critical Point (?, ?) is a critical point of 𝑓(?, ?) if 𝑓 ? (?, ?) = 0 and 𝑓 ? (?, ?) = 0 Relative maximum and minimum values can occur only at critical points .
Ex) Find all critical points: 𝑓(?, ?) = 2? 2 + ? 2 + 2?? + 4? + 2? + 5 Second-Derivative Test for Functions 𝑓( ? , ?) : The D-Test If (?, ?) is a critical point of the function 𝑓 and if ? is defined by ? = 𝑓 ?? (?, ?) ∙ 𝑓 ?? (?, ?) − [𝑓 ?? (?, ?)] 2 Then 𝑓 at the point (?, ?) has a : i. relative maximum if ? > 0 and 𝑓 ?? (?, ?) < 0 ii. relative minimum if ? > 0 and 𝑓 ?? (?, ?) > 0 iii. Saddle point if ? < 0
Ex) Find relative extreme values of 𝑓(?, ?) = 2?? − 2?2− 3?2+ 4? − 12? + 5
#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? − ?? + 6 1 Thousand dollars, find the quantities and the prices of the two products that maximize profit. Also find the maximum profit.

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