In this paper, we study the class of quintic (α, β)-metrics. We show that every weakly Landsberg 5-th root (α, β)-metrics has vanishing Scurvature. Using it, we prove that a quintic (α, β)-metric is a weakly Landsberg metric if and only if it is a Berwald metric. Then, we show that a quintic (α, β)-metric satisfies Ξ = 0 if and only if S = 0.
