Consider the following spectrum management problem. Suppose that there are n frequency tones available for the use of communication. The background noises are assumed to be additive Gaussian, and the noise at tone i is i>0,i=1,2,,n. Suppose that the user wishes to find an allocation of his/her communication powers over the n frequency tones, and denote xi to be the power allocated to tone i,i=1,2,,n. According to the information theory, the information rate on frequency tone i is ln(1+ixi),i=1,2,,n. Therefore the total information rate is i=1nln(1+ixi). The problem of maximizing the total information rate is to find the optimal power allocation x1,x2,,xn for the following optimization problem (SMP)maxs.t.xi0,i=1,2,,nf(x)=i=1nln(1+ixi)i=1nxiP where i>0,i=1,2,,n, and P>0 are the parameters of this model, and n is the dimension of the model. Solve (SMP) using the KKT optimality condition