quiz2 - (a Determine the critical point(s of f(b Show that...

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Spring 2006 Math 332 - Real Analysis II Quiz 2, 5/4/2006 1. Let f : R n R be a C 2 function; x o , u R n be fixed. Consider h : R R defined as h ( t ) = f ( x 0 + ut ). Calculate Dh (0) and D 2 h (0) in terms of the partial derivatives of f . Show your work in full detail . 2. We now finish the example we did not finish during the class; we investigate the nature of the critical point(s) of f ( x, y ) = x 2 + 2 xy + y 2 + 6. One easy way to do this is by observing that f ( x, y ) = ( x + y ) 2 +6. We choose a longer approach which would be valid for similar cases.
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Unformatted text preview: (a) Determine the critical point(s) of f . (b) Show that the bilinear form D 2 f ( a ) is semi-definite. Use directly the definition of positive/negative (semi)-definiteness of bilinear forms. Do not use the criteria that we did not prove in the class. (c) Along the direction vectors for which the strict definiteness is bro-ken, investigate if f increases or decreases. Ferit ¨ Ozt¨urk, Bo˘gazi¸ci University 2006...
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