Engineering Calculus Notes 378

Engineering Calculus Notes 378 - We have f x ( x,y ) = 3 x...

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366 CHAPTER 3. REAL-VALUED FUNCTIONS: DIFFERENTIATION the function has a local minimum at ( 1 2 , 1 2 ) . As another example, f ( x,y ) = 5 x 2 + 26 xy + 5 y 2 36 x 36 y + 12 has f x ( x,y ) = 10 x + 26 y 36 f y ( x,y ) = 26 x + 10 y 36 so the sole critical point is (1 , 1); the second partials are f xx ( x,y ) = 10 f xy ( x,y ) = 26 f yy ( x,y ) = 10 so the discriminant is Δ 2 (1 , 1) = (10) · (10) (26) 2 < 0 and the function has a saddle point at (1 , 1). Finally, consider f ( x,y ) = x 3 y 3 + 3 x 2 + 3 y.
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Unformatted text preview: We have f x ( x,y ) = 3 x 2 + 6 x = 3 x ( x + 2) f y ( x,y ) = 3 y 2 + 3 and these both vanish when x = 0 or 2 and y = 1, yielding four critical points. The second partials are f xx ( x,y ) = 6 x + 6 f xy ( x,y ) = 0 f yy ( x,y ) = 6 y so the discriminant is 2 ( x,y ) = (6 x + 6)( 6 y ) = 36( x + 1) y....
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This note was uploaded on 10/20/2011 for the course MAC 2311 taught by Professor All during the Fall '08 term at University of Florida.

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