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• Notes

Use Lagrange multipliers to Find three positive numbers whose sum is 100 and whose product is a maximum.

$f(x, y, z)=x y z$

$g(x, y, z)=x+y+z=100$

$\nabla f(x, y, z)=\lambda \nabla g(x, y, z)$

$\rightarrow x$

$y z=\lambda * 1 \rightarrow (1)$

$\rightarrow y$

$x z=\lambda * 1 \rightarrow (2)$

$\rightarrow z$

$yx =\lambda * 1 \rightarrow (3)$

$x+y+z=100 \rightarrow (4)$

$x, y, z, \lambda$

$\lambda=z y=x z=y x$

$x=y$

$y=z$

$x=y=z$

$3 x=100$

$x=\frac{100}{3}$

$\frac{100}{3}, \frac{100}{3}, \frac{100}{3}$

$\lambda=\frac{10000}{9}$