若\(\left\{ {{a_n}} \right\}\)是等差数列，首项\({a_1} > 0\)，\({a_{2021}} + {a_{2022}} > 0\)，\({a_{2021}}{a_{2022}} < 0\)，则使前\(n\)项和\({S_n} > 0\)成立的最大自然数\(n\)是（       ）

由\({a_1} > 0\)，\({a_{2021}}{a_{2022}} < 0\)，

则在等差数列中\({a_{2021}} > 0,{a_{2022}} < 0,\)

又\({a_{2021}} + {a_{2022}} > 0\)

可得：\({S_{4043}} = \frac{{{a_1} + {a_{4043}}}}{2} \times 4043 = 4043{a_{2022}} < 0\)，

\({S_{4042}} = \frac{{{a_1} + {a_{4042}}}}{2} \times 4042 = 2021({a_{2021}} + {a_{2022}}) > 0\)，

所以使前\(n\)项和\({S_n} > 0\)成立的最大自然数\(n\)是\(4042\).

故选：B

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