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

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

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

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

$$\rightarrow x$$

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

$$\rightarrow y$$

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

$$\rightarrow z$$

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

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

$$x, y, z, \lambda$$

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

$$x=y$$

$$y=z$$

$$x=y=z$$

$$3 x=100$$

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

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

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

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