Joint probability of two random variables both following poisson distribution

I have this joint probability question I'm having trouble with. If X, Y follow a poisson distribution with Poisson($\lambda$) and X and Y are independent. Show that: $$1 \le \frac{VAR(max{X,Y})}{\lambda} \le 2$$

I know that variance of a random variable following a poisson distribution is $\lambda$. Any help would be greatly appreciated!

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hint: the cdf for the maximum is just the product of the cdf’s for the individual rv’s. Just multiply the two CDF's and I am sure you can calculate the variance of $max(X,Y)$. –  jay-sun Nov 20 '12 at 19:10
@jay: I doubt that. Can you? –  Robert Israel Nov 20 '12 at 19:48
@RobertIsrael: I see your point here. –  jay-sun Nov 20 '12 at 21:35

In fact, it is always true that for two random variables $X$ and $Y$ (independent or not), $\text{Var}(\max(X,Y)) + \text{Var}(\min(X,Y)) \le \text{Var}(X) + \text{Var}(Y)$. This gives you the upper bound. I don't see an easy way to get the lower bound.