# SurCalCh7Sec3 - f Example 1#8 in 7.3 Find the relative...

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Optimizing Functions of Several Variables The point (a,b) is a critical point of if and ) , ( y x f () 0 , = b a f x () 0 , = b a f y To find the maxima, minima, and saddle points of a function we first find the critical points as follows: ) , ( y x f 1) Set the partial derivatives equal to zero 2) Solve the resulting system of equations Next, we apply the following test (called the D-test) If (a,b) is a critical point of the function , then for D defined by : ) , ( y x f () () 2 ] , [ , ) , ( b a f b a f b a f D xy yy xx = a) has a relative maximum at (a,b) if D>0 and f () 0 , < b a f xx b) has a relative minimum at (a,b) if D>0 and f () 0 , > b a f xx c) has a saddle point at (a,b) if D<0
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Unformatted text preview: f Example 1 (#8 in 7.3) Find the relative extreme values of each function 10 7 5 3 2 5 ) , ( 2 2 + − + − − = y x y x xy y x f Example 2 (#16, 7.3) Find the relative extreme values of y x y x 6 3 2 3 + − − Example 3 (# 24, 7.3) In a laboratory test the combined antibiotic effect of x milligrams of medicine A and y milligrams of medicine B is given by the function (for , ). Find the amounts of the two medicines that maximize the antibiotic effect. y x y x xy y x f 60 110 2 ) , ( 2 2 + + − − = 55 ≤ ≤ x 60 ≤ ≤ y...
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