Question

Suppose that f(x,y)=xy, with the constraint that x and y are constrained to sum to 1. That is, x + y = 1. 

Given this constraint, which of the following functions of x is equivalent to the original function f(x,y)=xy? 

$$ \begin{aligned} &\tilde{f}(x)=1-x \\ &\tilde{f}(x)=x-x^{2} \\ &\tilde{f}(x)=x+x^{2} \\ &\widetilde{f}(x)=x^{2} \end{aligned} $$

The langrange method can also be used to solve this constrained maximization problem.

The langrangian for this constrained maximization problem is _______ 

Which of the following are the first order conditions for a critical point for the langranian function L ? Check all that apply.

$$ \begin{aligned} &\frac{d L}{d \lambda}=1-x=0 \\ &\frac{d L}{d x}=y-\lambda=0 \\ &\frac{d L}{d \lambda}=1-x-y=0 \\ &\frac{d L}{d y}=x-\lambda=0 \\ &\frac{d L}{d y}=y \lambda=0 \end{aligned} $$

Answer 1

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