= Solution
A <Bloch state> in a crystal experiences a periodic lattice potential. Its band <dispersion relation> $\epsilon(k)$ encodes the lattice response, so an <effective mass> allows a wavepacket's acceleration under a weak applied <force> to be described without explicitly following the microscopic periodic <forces>.
In one dimension the wavepacket <group velocity> and semiclassical equation for <crystal momentum> are
$$
v_g=\frac1\hbar\frac{d\epsilon}{dk},\qquad \hbar\dot k=F.
$$
Differentiating the first equation with respect to time gives
$$
a=\frac1\hbar\frac{d^2\epsilon}{dk^2}\dot k
=\frac{F}{\hbar^2}\frac{d^2\epsilon}{dk^2}.
$$
Defining $F=m^*a$ yields
$$
\boxed{m^*=\hbar^2\left(\frac{d^2\epsilon}{dk^2}\right)^{-1}}.
$$
This <band effective mass> is controlled by <band curvature>, not the slope. A band minimum has positive $m^*$; a band maximum has negative <Electron> $m^*$ and is often more conveniently described with positive-mass <electron holes>. In three dimensions the inverse <effective mass> is generally a tensor, $(m^{*-1})_{ij}=\hbar^{-2}\partial_{k_i}\partial_{k_j}\epsilon$. The scalar formula assumes motion along a specified direction in an isolated band, weak fields and negligible interband transitions; the bare <Electron> <mass> has not physically changed.
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