We propose a non-invasive and cost-effective method to automatically detect dementia by utilizing solely speech audio data. We extract paralinguistic features for a short speech segment and use Gated Convolutional Neural Networks (GCNN) to classify it into dementia or healthy. We evaluate our method on the Pitt Corpus and on our own dataset, the PROMPT Database. Our method yields the accuracy of 73.1% on the Pitt Corpus using an average of 114 seconds of speech data. In the PROMPT Database, our method yields the accuracy of 74.7% using 4 seconds of speech data and it improves to 80.8% when we use all the patient's speech data. Furthermore, we evaluate our method on a three-class classification problem in which we included the Mild Cognitive Impairment (MCI) class and achieved the accuracy of 60.6% with 40 seconds of speech data.