If x, y, z in mathbb{R}^+ such that x + y + z | vedclass

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(D) We are given $x + y + z = 4$,where $x, y, z > 0$. We want to maximize $xyz^2$.Using the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality for

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$4$

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(D) We are given $x + y + z = 4$,where $x, y, z > 0$. We want to maximize $xyz^2$.Using the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality for four positive numbers $x, y, z/2, z/2$:$\frac{x + y + z/2 + z/2}{4} \geq \sqrt[4]{x \cdot y \cdot \frac{z}{2} \cdot \frac{z}{2}}$Substituting $x + y + z = 4$:$\frac{4}{4} \geq \sqrt[4]{\frac{xyz^2}{4}}$$1 \geq \sqrt[4]{\frac{xyz^2}{4}}$Raising both sides to the power of $4$:$1 \geq \frac{xyz^2}{4}$$xyz^2 \leq 4$Thus,the maximum value is $4$.

If $x, y, z \in \mathbb{R}^+$ such that $x + y + z = 4$,then the maximum possible value of $xyz^2$ is

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