MATA33H3 Lecture Notes - Lecture 22: Lagrange Multiplier, Ellipse
17.7 – Lagrange Multipliers (con’t)
Review from last lecture:
Definition:
o is called the Lagrange Multiplier.
o A technique/method to find and optimize local extrema of a function of > 2 variables
where the function is subject to > 1 constraint equations
o Max and Min of f (x,y,z) are subjects to one constraint g(x,y,z) = K
1. Method of lagrange multipliers for one constraint
Overview Example:
Let z = f ( x , y ) = 5x + 2y which is subject to constraint of 5x^2 + 2y^2 = 14
a) E = { ( x , y ) | 5x^2 + 2y^2 = 14 }
**we only consider points that are on the ellipse**
To find max and min of f (x,y,z) subject to one constraint g(x,y,z) = K
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To find max and min of f (x,y,z) subject to one constraint g(x,y,z) = k. Overview example: e = { ( x , y ) | 5x^2 + 2y^2 = 14 } Let z = f ( x , y ) = 5x + 2y which is subject to constraint of 5x^2 + 2y^2 = 14. **we only consider points that are on the ellipse*: find all constrained critical points. Constrained function : g(x,y) = 5x^2 + 2y^2 14. + hz: largest of these values = max of f, smallest of these values = min of f. Compute the max and min of the function u = f (x,y,z) Subject to constraints g ( x , y , z ) = k h ( x , y , z ) = l. Must solve: f ( x , y , z ) = x^2 + 2y^2 + 3yz + 4z^2 x + y + z = 1.
