Slide_08 - MA1104 Multivariable Calculus Lecture 8 Dr KU...

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Unformatted text preview: MA1104 Multivariable Calculus Lecture 8 Dr KU Cheng Yeaw Monday Feb 9, 2009 Recap Last lecture, • we learned about the significance of the gradient vector O f . (i) O f ( x ,y ) is perpendicular to the level curve f ( x,y ) = k at ( x ,y ) , O F ( x ,y ,z ) is perpendicular to the tangent plane to the level surface F ( x,y,z ) = k at ( x ,y ,z ) . (remember we need the assumption that f and F are differentiable) (ii) O f points in the direction of maximum rate of increase • we use the Second Derivative Test to find local maximum/minimum Overview In this lecture, • we will learn how to find absolute extremum on closed bounded region • we want to optimize f ( x,y ) subject to the constraint g ( x,y ) = k . This method is called the Method of Lagrange Multipliers. Absolute Maximum/Minimum - Extreme Value Theorem So far we have been interested in local maximum/minimum and saddle points. What about absolute maximum/minimum points? Firstly, they do not always exist. For instance, f ( x,y ) = x 3- 2 y 2- 2 y 4 + 3 x 2 y has a local maximum value f (- 2 , 1) = 16 (see this example in Lecture Slide 7). This value cannot be an absolute maximum as f achieves greater value at other points, say f (4 , 0) = 4 3 > 16 . Nevertheless, if we restrict the domain of our function to a closed and bounded region, then absolute maximum and absolute minimum always exist! This is the content of the Extreme Value Theorem ....
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Slide_08 - MA1104 Multivariable Calculus Lecture 8 Dr KU...

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