Take mx′′+cx′+kx=F0cos(ωt).mx″+cx′+kx=F0cos⁡(ωt). Fix c>0,c>0, k>0,k>0, and F0>0.F0>0. Consider the function C(ω).C(ω). For what values of mm (solve in terms of c,c, k,k, and F0F0) will there be no practical resonance (that is, for what values of mm is there no maximum of C(ω)C(ω) for ω>0ω>0)?

Question

Take mx′′+cx′+kx=F0cos(ωt).mx″+cx′+kx=F0cos⁡(ωt). Fix c>0,c>0, k>0,k>0, and F0>0.F0>0. Consider the function C(ω).C(ω). For what values of mm (solve in terms of c,c, k,k, and F0F0) will there be no practical resonance (that is, for what values of mm is there no maximum of C(ω)C(ω) for ω>0ω>0)?