Yes, you are required to file your tax return by October 15 if you filed for a six-month extension. A tax extension, filed using IRS Form 4868, is simply a notification to the IRS that you were not able to file a complete return by the deadline; you do not have to provide a reason.
Remember that this is only an extension of time to file, not to pay. If you owe any tax, payment is still due by April 15 (or the next business day, if the 15th falls on a holiday or weekend). In order to avoid late-filing penalties, you must file your tax return prior to your extended deadline of October 15 (or the next business day, if the 15th falls on a holiday or weekend).