= Solution
A <Lipschitz gradient> with constant $\gamma>0$ gives the <descent lemma>
$$
f(x+\alpha p)\leq f(x)+\alpha g^Tp+\frac\gamma2\alpha^2\|p\|_2^2.
$$
For $\alpha>0$, this upper bound is at most $f(x)+\alpha\beta g^Tp$ whenever $\tfrac\gamma2\alpha\|p\|_2^2\leq(\beta-1)g^Tp$. Since $p$ is a <descent direction>, the right side is positive. Thus the <Armijo condition> holds throughout
$$
\boxed{0\leq\alpha\leq\alpha_{\max}=\frac{2(\beta-1)g^Tp}{\gamma\|p\|_2^2}.}
$$
At $\alpha=0$ it holds by equality. If the gradient Lipschitz constant can be taken as zero, the objective is affine and every positive step along a descent direction passes this sufficient-decrease test.
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