The author proves that if \(q\) is odd and congruent to 1 modulo 3, then the polynomial \(f(x)=x^{1+(q-1)/3}+ax\) \((a\neq 0)\) is not a permutation polynomial over any field \(\mathbb F_{q^r}\) \((r\geq 2)\). Thereby a question raised by \textit{L. Carlitz} [Bull. Am. Math. Soc. 68, 120--122 (1962); Zbl 0217.33003)] receives a partial answer. In addition, it is shown that if \(p\) is an odd prime, \(1(k-1,p-1)(k-1)\), then \(f(x)=x^k+ax\) \((a\neq 0)\) is not a permutation polynomial over \(\mathbb F_p\). This result follows from Newton's identities of power sums as used by \textit{L. E. Dickson} [Linear groups. 2nd ed. New York: Dover Publications (1958; Zbl 0082.24901), {\S}{\S} 74, 75 and 84].