minimum value of $x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}$

Question

If $x,y,z,t\in N.$ Then maximum and minimum value of $$x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}$$

what i try

$$\frac{(x+x+\cdots \text{x times} )+(y+y+\cdots \text{y times})+(z+z+\cdots \text{z times})+(t+t+\cdots \text{t times})}{x+y+z+t}\geq \bigg(x^x\cdot y^y\cdot z^z \cdot t^t\bigg)^{\frac{1}{x+y+z+t}}$$

$$x^{\frac{x}{x+y+z+t}}\cdot y^{\frac{y}{x+y+z+t}}\cdot z^{\frac{z}{x+y+z+t}}\cdot t^{\frac{t}{x+y+z+t}}\leq \frac{x^2+y^2+z^2+t^2}{x+y+z+t}$$

how do i find minimum value of expression help me please

Answer 1

The minimum is $1$ and occurs for $x=y=z=t=1.$

The maximal value does not exist. Tty $x=y=z=t\rightarrow+\infty.$

Answer 2

Hint: Let $x=y=z=t$, then the expression becomes $t$, which can be made equal to any $t\in \mathbb N$… now what can you conclude on minimum or maximum (if it exists)?

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P.S. It seems the OP really wanted to show the expression is $\geqslant \frac14(x+y+z+t)$. For this, from homogeneity (we may extend the domain to positive reals as $\mathbb N$ is a subset), we may assume $x+y+z+t=4$, so that it is enough to show: $$\prod_{x, y, z, t} x^x \geqslant 1 \iff \sum_{x, y, z, t} x \log x \geqslant 0$$ which follows directly from Jensen's inequality on the convex function being summed.