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

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(D) Given $x + y + z = 4$ where $x, y, z \in R^+$.We want to maximize $xyz^2 = x \cdot y \cdot \frac{z}{2} \cdot \frac{z}{2} \cdot 4$.By the Arithmeti

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$4$

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(D) Given $x + y + z = 4$ where $x, y, z \in R^+$.We want to maximize $xyz^2 = x \cdot y \cdot \frac{z}{2} \cdot \frac{z}{2} \cdot 4$.By the Arithmetic Mean-Geometric Mean Inequality ($AM$-$GM$),for positive real numbers $x, y, \frac{z}{2}, \frac{z}{2}$:$\frac{x + y + \frac{z}{2} + \frac{z}{2}}{4} \geq \sqrt[4]{x \cdot y \cdot \frac{z}{2} \cdot \frac{z}{2}}$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 $x, y, z \in R^+$ such that $x + y + z = 4$,then the maximum possible value of $xyz^2$ is -

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