In this correspondence paper, the problem of noncoherent detection of a sinusoidal carrier is considered in the presence of Gaussian noise. The convexity properties of the detection probability are characterized with respect to the signal-to-noise ratio (SNR). It is proved that the detection probability is a strictly concave function of SNR when the false alarm probability alpha satisfies alpha > e(-2), and it is first a strictly convex function and then a strictly concave function of SNR for alpha < e(-2). In addition, optimal power allocation strategies are derived under average and peak power constraints. It is shown that ON-OFF signaling can be optimal for alpha < e(-2) depending on the power constraints, whereas transmission at a constant power level that is equal to the average power limit is optimal in all other cases.