The authors consider the problem of minimizing the function \(h(x):=\max\{f(x,\tau)\mid \tau \in T\}\), where \(x\in R^{p}\), \(T\) is a compact metric space and the function \(f(\cdot ,\tau)\) is \(C^{2}\). Second-order necessary optimality conditions are obtained by using second-order directional derivatives of Chaney and Ben-Tal/Zowe of \(h\). These conditions improve an earlier result of \textit{H. Kawasaki} [Appl. Math. Optimization 26, No. 2, 195-220 (1992; Zbl 0777.90058)].