If x, y, z in R^+ such that x + y + z = 4,the | vedclass

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(D) To find the maximum value of $xyz^2$ given $x + y + z = 4$ where $x, y, z \in R^+$,we use the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequalit

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$4$

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(D) To find the maximum value of $xyz^2$ given $x + y + z = 4$ where $x, y, z \in R^+$,we use the Arithmetic Mean-Geometric Mean ($AM$-$GM$) inequality.We want to maximize the product $x \cdot y \cdot (z/2) \cdot (z/2)$.According to the $AM$-$GM$ inequality,for positive real numbers,the arithmetic mean is greater than or equal to the geometric mean:$\frac{x + y + z/2 + z/2}{4} \geq \sqrt[4]{x \cdot y \cdot (z/2) \cdot (z/2)}$Substituting the given sum $x + y + z = 4$:$\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 possible value of $xyz^2$ is $4$.

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

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