This paper proposes a Convolutional Neural Network (CNN) inspired by Multitask Learning (MTL) and based on speech features trained under the joint supervision of softmax loss and center loss, a powerful metric learning strategy, for the recognition of emotion in speech. Speech features such as Spectrograms and Mel-frequency Cepstral Coefficient s (MFCCs) help retain emotion-related low-level characteristics in speech. We experimented with several Deep Neural Network (DNN) architectures that take in speech features as input and trained them under both softmax and center loss, which resulted in highly discriminative features ideal for Speech Emotion Recognition (SER). Our networks also employ a regularizing effect by simultaneously performing the auxiliary task of reconstructing the input speech features. This sharing of representations among related tasks enables our network to better generalize the original task of SER. Some of our proposed networks contain far fewer parameters when compared to state-of-the-art architectures.