math16B_suppl_notes_2 - Math 16B Spring 2011 Supplementary...

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Math 16B – Spring 2011 – Supplementary Notes 2 Second-Derivative Test To understand what is behind the second-derivative test for functions of two variables, we shall start by looking at the simplest nontrivial example, that of a polynomial of degree 2. First the test will be stated. Let the function f ( x,y ) have a critical point at ( a,b ). The second-derivative test involves the function D f ( x,y ) = ± 2 f ∂x 2 ²± 2 f ∂y 2 ² - ± 2 f ∂x∂y ² 2 , and it applies when D f ( a,b ) 6 = 0. Note that if D f ( a,b ) > 0 then 2 f ∂x 2 ( a,b ) and 2 f ∂y 2 ( a,b ) must have the same sign. The test distinguishes three cases: (I) If D f ( a,b ) > 0 and 2 f ∂x 2 ( a,b ) > 0 (equivalently 2 f ∂y 2 ( a,b ) > 0), then ( a,b ) is a relative minimum of f . (II) If D f ( a,b ) > 0 and 2 f ∂x 2 ( a,b ) < 0 (equivalently 2 f ∂y 2 ( a,b ) < 0), then ( a,b ) is a relative maximum of f . (III) If D f ( a,b ) < 0 then ( a,b ) is a saddle point of f (neither a relative maximum nor a relative minimum). Now we look at the simple example f ( x,y ) = αx 2 +2 βxy + γy 2 , where α,β,γ are constants, not all 0. This quadratic polynomial has a critical point at (0
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math16B_suppl_notes_2 - Math 16B Spring 2011 Supplementary...

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