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I am studying past exam answers for a remote econ math exam, but I don't understand the given solution for this question:

$$ \begin{array}{ll} \text{Maximize}& z(x,y) = y e^{5x} \\ \text{Subject to}& ax + by = 8 \\ \text{Where} & x\geq 0~\text{and}~ y \geq 0 \end{array} $$ The maximum location of this problem is $(x,y) = \left(\frac{1}{5}, 1\right)$. Determine $a$ and $b$.

solution to problem

In the solution they use "first order condition"":

$5y\cdot e^{5x} / e^{5x} = a/b$

$ax + by = 9$

To come to $5y = a/b$

and $ax + by = 8$, see image.

But I don't understand where this "first order condition" comes from and how they come to these conclusions, could somebody explain it to me please? I searched online but can't find anything.

CrSb0001
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noblearistocrat
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1 Answers1

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Form the Lagrangian

$$ L(x,y,\lambda) = y e^{5x}-\lambda(a x+b y -8) $$

then from the stationary conditions $\nabla L = \cases{5ye^{5x}-\lambda a = 0\\ e^{5x} - \lambda b = 0\\ a x+b y -8 = 0}$

now eliminating $\lambda$ we obtain

$$ \lambda = \frac{5ye^{5x}}{a} = \frac{e^{5x}}{b}\Rightarrow 5y = \frac ab $$

Cesareo
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