Solve an Ellipse Product Min Problem with LaGrange

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To solve the problem of minimizing the product xy on the ellipse defined by 8x^2 + 2y^2 = 16, LaGrange multipliers can be applied. The objective function to minimize is xy, while the constraint is the equation of the ellipse. The first derivative of the objective function must equal zero for a minimum, and the second derivative must be positive. The ellipse serves as the constraint in this optimization problem. Understanding these concepts is crucial for finding the minimum points effectively.
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Hello, the problem that i can't figure out says: To Use LaGrange multipliers to find all points on the ellipse 8x^2 + 2y^2 = 16 at which the product xy is a minimum? Any hints appreciated.
 
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What have you done so far?
 
I'm just not sure what the "which the XY product is a minimum" is really asking me.
 
xy is your objective function, the one you're trying to minimize.
 
to have a minimum of a function the 1st derivative must equal 0 and the second derivative must be greater than 0.

does that help?
 
ah ok, and my ellipse is the constraint then?
 
Looks that way.
 

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