= Solution
First convert from a <spectral energy density> per unit <frequency> to one per unit <wavelength>, including the <Jacobian matrix and determinant>:
$$
U_\lambda=U_\nu\left|\frac{d\nu}{d\lambda}\right|=\frac{8\pi hc}{\lambda^5}\frac1{e^{hc/(\lambda k_BT)}-1}.
$$
Set $dU_\lambda/d\lambda=0$ and define $x=hc/(\lambda k_BT)$. The nonzero maximum satisfies $5(1-e^{-x})=x$, with numerical root $x=4.96511$. Consequently
$$
\boxed{\lambda_{\max}T=\frac{hc}{4.96511k_B}=2.89777\times10^{-3}\,\mathrm{m\,K}.}
$$
This is <Wien's displacement law>. One must maximize the wavelength density, rather than find the frequency maximum and merely put $\lambda=c/\nu_{\max}$: the <frequency and wavelength peaks of Planck's law> are different because a spectral density changes under a change of coordinate.
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