When the function has a limit, to express the "constrained" max or min you set it equal to the upper or lower level, respectively. Graphically, the limit is represented by a line that intersects with the contours of the function line. The max lies on this line because anything bellow is not reaching full potential by not hitting the limit, and everything above is disallowed because it exceeds the limit. Lagrange multipliers can also be applied to a constrained eqn to optimize it. Local max.s occur at a particular point that are greater than all immediate points and meets the requirements of the constrainment, while the global max this is true for all points of the function, vice versa for min.s. The optimization is found by plugging the function into three specific eqns. The Legrange multiplier used in these eqns is defined by the optim value of a funtion as the constrained value is increased by one unit. The Legrange multiplier is the rate of change of the increase in a function per unit. Furthermore a constrained function can be optimized by plugging it into the Legrange function and determining the critical points.
Challenges
Why do you set the limit equal to the limit instead of it having be less than or equal to? If there is a limit, like the 378,000 used in the book, how can that possibly represent a line as opposed to a point? so the legrange multiplier is not like e right, in that it is a special number? is it a special ratio? i dont understand it's significance.
Reflections
Because the Legrange multiplier represents a rate of change, is it closely connected with the derivative function? Is it the deriv.?
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