The extreme shear stresses with the corresponding planes, which play the important role in some well known yielding criteria, are established. In the first part the extreme shear stresses and the corresponding planes are obtained with the help of the elementary inequalities, strictly avoiding any usage of differential calculus, so the undergraduate students of technical studies can easily follow the proof. In the second part the characters of the stationary points obtained by the method of Lagrangian multipliers are explained in an easily understandable way using the some well known computer graphics tools like gnuplot, SAGE, R, MATLAB and Mathematica.