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quiz6soln - Solution Let f x,y,z = x-j 2 y-k 2 z-l 2-r 2...

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MATH 231 / Spring 2008 March 12, 2008 Name: Key 1. (3 points) Suppose the second partial derivatives of f are continuous on a disk with center ( a, b ) and suppose that f x ( a, b ) = f y ( a, b ) = 0. What does the second derivative test allow us to infer under such hypotheses? Solution: Let D = f xx ( a, b ) f yy ( a, b ) - [ f xy ( a, b )] 2 . Then (a) If D > 0 and f xx ( a, b ) > 0, then f ( a, b ) is a local minimum. (b) If D > 0 and f xx ( a, b ) < 0, then f ( a, b ) is a local maximum. (c) If D < 0, then f ( a, b ) is a saddle point. 2. (7 points) Given a point ( j, k, l ) and a positive real number r , show that every normal line to the sphere ( x - j ) 2 + ( y - k ) 2 + ( z - l ) 2 = r 2 passes through the center of the sphere.
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Unformatted text preview: Solution: Let f ( x,y,z ) = ( x-j ) 2 +( y-k ) 2 +( z-l ) 2-r 2 . Then we know that ∇ f = 2 h x-j,y-k,z-l i . Letting ( x ,y ,z ) be a general point on the sphere, we now have what we need to compute the normal line: * x y z + = ∇ f ( x ,y ,z ) t + * x y z + Thus, we get the following three equations: x = ( x-j ) t + x y = ( y-k ) t + y z = ( z-l ) t + z Evaluating this line at the point t =-1 gives us ( x,y,z ) = ( j,k,l ), the center of the circle. 1...
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