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Unformatted text preview: m at . Second Partials Test If D(a,b) > 0 and > 0 THEN Relative Minimum If D(a,b) > 0 and < 0 THEN Relative Maximum If D(a,b) < 0 THEN Saddle Point
If D(a,b) = 0 THEN Test Inconclusive Extreme Value Theorem
assumes an absolute extremum on any closed bounded set S. All absolute extremum must occur either on the
boundary of S or at a critical point in the interior of S. Regression
Least Squares Criteria: Let and So we have x
Inputting the values for x and y: Solve this in Mathematica and show the scatterplot and the regression line with the regression equation Section 12.9 Lagrange Multipliers
IF maximizes or minimizes the function subject to the constraint And if
Then for some constant In other words, constrained extrema can only occur when the gradient vectors and are parallel. Proof:
Let represent the point Let the point
Let and let with the constraint Curve C: lie on the curve maximize or minimize the function subject to the constraint We can think of Curve C as a level curve for the function
Thus we know and ( ie: ) is orthogonal to the curve C at Let be a parameterization of the curve C such that We are trying to find the extrema of the function
This is similar in nature to the derivation of
restricted to the line in the direction of the vector
(a curve instead of a line) with the constraint Curve C: Thus
We know the extrema happen when Since the extrema happen at or we have: is orthogonal to . which we did above. Remember our domain was
. Now we have a function that has a domain restriction of So we are looking for the extrema of Thus . or in other words orthogonal to the curve C at Earlier we showed: is orthogonal to the curve C at Finally this implies and are parallel. . We will use this theorem to find extrema. We will start with
to get a list of possible extrema. This equation does not guarantee you have found an extrema but
only gives you the list of possible extrema. We have no simple test to distinguish maxima and minima as we do for un...
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This note was uploaded on 03/05/2014 for the course MATH 101c taught by Professor Loukianoff,v during the Spring '08 term at Ohlone.
- Spring '08