Exact results on the general higher-order cusp relations involving the second derivative of electron density are derived for atoms and molecules including highly excited states wherein the spherical average of the derivative of the wave function is zero at a nucleus. The curvature of the electron density at the nucleus is found to be a direct measure of the extent of excitation. An expression relating the one-electron eigenvalues to quantities at the cusp and the atomic number is also obtained. Numerical. tests on several excited-state atoms are presented.