Explicit expressions of the direction n and the stationary values (maximum, minimum and saddle point) of Young's modulus E(n) for orthotropic, tetragonal, trigonal, hexagonal and cubic materials are presented. For the shear modulus G(n, m), explicit expressions of the extrema (maximum and minimum) and the two mutually orthogonal unit vectors n, m are given for cubic and hexagonal materials. We also present a general procedure for computing the extrema of G(n, m) for more general anisotropic elastic materials. It is shown that Young's modulus E(n) can be made as large as we wish for certain n without assuming that the elastic compliance s11, s22 or s33 is very small. As to the shear modulus G(n,m), it can be made as large as we wish for certain n and m without assuming that any one of the elastic compliance sαβ is very small. We also show that Young's modulus E(n) can be independent of n for orthotropic and hexagonal materials while the shear modulus G(n, m) can be independent of n and m for hexagonal materials.