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Hi i was wondering if you can help me with this question.
Among all pairs of positive numbers whose product equals 25, find the pair with the smallest sum. Formulate this as a two-variable optimization problem with one constraint and solve it in two ways. First, by reduction to one variable and, second, by Lagrange multipliers.
Take x to be one number - then 25/x is the other number. The answer is finding a minimum to x+25/x.
By applying the principle of Lagrange multipliers, we can say that we are trying to minimize xy where x*y=25. What we have is f(x,y)=xy and g(x,y)=xy-25.
The Lagrange formula is then L(x,y,s) = f(x,y)-sg(x,y).
Set the parial derivatives of x, y, and s all equal to 0 and solve.
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