= Solution
For <Planck law> expressed per unit <frequency>, put $x=h\nu/(k_BT)$. At fixed <temperature>, the frequency dependence is $B_\nu\propto T^3x^3/(e^x-1)$. Differentiation gives
$$
\frac{d}{dx}\frac{x^3}{e^x-1}
=\frac{x^2[3(e^x-1)-xe^x]}{(e^x-1)^2}.
$$
The positive interior maximum therefore satisfies $3(1-e^{-x_*})=x_*$, with $x_*=2.821439\ldots$ independent of <temperature>. The zero solution of the rearranged equation is a boundary, not the interior maximum. Hence the <frequency form of Wien's displacement law> is
$$
\boxed{\frac{\nu_{\max}}{T}=\frac{x_*k_B}{h}
\simeq5.88\times10^{10}\,\mathrm{Hz\,K^{-1}}.}
$$
This frequency peak is not $c$ divided by the peak wavelength of $B_\lambda$: the two spectral densities differ by the frequency-to-wavelength Jacobian.
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