Utility maximization problem : free product vs priced product? Here is the original question :


  The ESPN cable TV network runs a major sports information site at www.espn.com. Most of the content is free, but ESPN has a premium membership (see its "Insider") available for a monthly or an annual fee. Similar, but fee-free, sports content can be found at the Web sites of CNN Sports Illustrated, sportsillustrated.cnn.com, and CBS Sports Line,www.cbssports.com. Since ESPN has put a price tag on some of its sports content, it implies that the utility of a premium membership cannot be found at a no-fee site and is therefore worth the price. Is this the case? Use the utility-maximization rule to justify your subscribing or not subscribing to the premium membership.


Which product will yield maximum utility?
Let say if one content is available for free and the other is priced but a utility of that cannot be found anywhere else.
Applying the law of Utility-maximization I get this
( Marginal utility of fee-free content)/(price of fee-free content)   ≠ ( Marginal utility of premium content)/(price of premium content )

So , I am thinking regardless of the price of a premium content the fee-free content’s marginal utility over price will never equal the marginal utility of premium content over the price of premium membership. So, why will one buy a premium product?
Am I correct on this?
I think there are some problems with the framing of your question. Some points... 

There are some assumptions that need to be added for a meaningful answer. 


Diminishing returns, budget constraint, free disposal etc. 
In terms of utility maximization, there needs to be a budget constraint. if you add one and seek to maximize utility, the consumer will hold $x_2=\frac{w}{p}$ (spend all their money on good 2, and then hold an infinite amount of good 1.


If you are thinking of something like an app, consider non monetary costs. They operate in the same way. Like the time value of playing a worse app. or movies, if you are given a choice to watch a terrible movie for free or pay to see a good movie, you may elect to pay. In which case time is a factor. You can't consume an infinite amount of terrible movies at time $t$.

In summary, you need to add more to the question. If a good is free, marginal utility has to diminish and $\frac{du}{dx}$ needs to equal $0$ at some point or supply has to be finite. Until at least that is satisfied, the consumer will spend all money on on $x_2$ and hold an infinite amount of $x_1$