The paper presents the design of a speed and flux observer for the sensorless induction motor (IM) drive. The design is based on the full-order model of the IM in the stationary reference frame. The proposed method consists of an observer with linear feedback that yields the motor fluxes and also estimates the speed using an adaptive law. Convergence conditions are formulated based on Lyapunov's nonlinear stability theory - the design of the feedback gains is done using Sylvester criterion. While this observer structure was discussed in the literature previously, the novelty of the method is given by the Sylvester criterion formulation - this uncovers interesting inequalities and offers insight into the design choices and stability margins of the observer. The method can be used to obtain the field orientation angle and speed needed in a sensorless IM drive. The theory is supported by simulations and experiments.