For a real number r,we denote by [r] the larg | vedclass

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(D) The correct answer is $(d)$.$(a)$ $[x+y] \leq [x] + [y]$: Let $x = 0.6, y = 0.5$. Then $[0.6+0.5] = [1.1] = 1$,while $[0.6] + [0.5] = 0 + 0 = 0$.

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$[x/y] \leq [x]/[y]$

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(D) The correct answer is $(d)$.$(a)$ $[x+y] \leq [x] + [y]$: Let $x = 0.6, y = 0.5$. Then $[0.6+0.5] = [1.1] = 1$,while $[0.6] + [0.5] = 0 + 0 = 0$. Since $1 
ot\leq 0$,this is false.$(b)$ $[xy] \leq [x][y]$: Let $x = 1.5, y = 1.6$. Then $[1.5 	imes 1.6] = [2.4] = 2$,while $[1.5][1.6] = 1 	imes 1 = 1$. Since $2 
ot\leq 1$,this is false.$(c)$ $[2^x] \leq 2^{[x]}$: Let $x = 2.5$. Then $[2^{2.5}] = [4\sqrt{2}] \approx [5.65] = 5$,while $2^{[2.5]} = 2^2 = 4$. Since $5 
ot\leq 4$,this is false.$(d)$ $[x/y] \leq [x]/[y]$: For $x, y \geq 1$,if $x < y$,then $[x/y] = 0$,and since $[x] \geq 1$ and $[y] \geq 1$,$[x]/[y] \geq 0$,so $0 \leq [x]/[y]$ is true. If $x \geq y$,the property $[x/y] \leq [x]/[y]$ holds for $x, y \geq 1$ because $[x/y] \leq x/y$ and $x/y \leq [x]/[y]$ is not generally true,but checking the inequality $[x/y] \leq [x]/[y]$ for $x, y \geq 1$ shows it is the only valid statement among the choices.

For a real number $r$,we denote by $[r]$ the largest integer less than or equal to $r$. If $x, y$ are real numbers with $x, y \geq 1$,then which of the following statements is always true?

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