Alternative maximum is not at the interior point where f exists and is not zero Lecture 9 Find maximums of y f x Tangent line approximation to y f x at x f x Slope f x y x x f x f x What is the direction 1 f x What is the normal Find maximum of f x y z Options for maximum values Point where f f 0 f x 1 f x what the tangent line is x x f x f x Means that the tangent plane is horizontal Point where f or f is not de ned Boundary point Tangent plane approximation f x y f x y x x f y y f What is the normal f f 1 Finding a minimum Step 1 Find f and f f 2x 3 y f x 2y Step 2 Check boundary curves Scenario 1 x 3 and y 2 f 3 y y 3y 19 Example Find a minimum for f x y x 3x xy y 1 in the rectangle x 3 and y 2 To minimize this take the rst derivative Now check y y 2 All of these points are bigger than 2 so they are not the minimum Scenario 2 x 3 and y 2 Scenario 3 y 2 and x 3 Scenario 4 y 2 and x 3 Plotting a collection of data points Data Least squares chooses a best line Best line y ax b Error D a b Minimize D a b overall a b D D These two lines intersect in a point This is your a and b
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