No, police have no "sworn duty" to investigate - you literally made this up out of your head.
“I, (state name), do solemnly swear or affirm that I will faithfully execute the duties of (state position) of the State of Texas, and will to the best of my ability preserve, protect and defend the Constitution and laws of the United States and of Texas, so help me God.”
"I do solemnly affirm that I will faithfully discharge the duties of the position of ______________________ according to the best of my ability and perform my duties in a manner consistent with the Constitution of the United States and the Constitution of the State of New York."
They're all like this.
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No, seeing a black guy watering flowers is not grounds for anything, even if they are someone else's flowers. I cannot believe I have to make this point.