Someone told me about a case where some people got lost in the woods, "the authorities" (not sure who, but let's assume they had a legal duty to rescue the people who got lost) could not find them, someone who the authorities would not necessarily have a reason to trust tracked one of the people's phones and gave their location to the authorities (while the lost people were presumably still alive), the authorities chose not to act on this data, the people died, and were later found at the location that was reported. Did "the authorities" do anything illegal? The person who told me about this did not clarify who "the authorities" are.
The authorities, if they are some form of law enforcement (sheriff, park rangers, etc.) have no duty to rescue unless they were the ones who put you in peril, for example, by detaining you in a patrol car and then parking it on train tracks while a train is coming.
If the authorities didn't cause them to get lost in the woods, the authorities have no legally enforceable duty to do anything. See generally, Castle Rock v. Gonzales, 545 U.S. 748 (2005) (holding that law enforcement discretion to act or not act is virtually absolute).
A different analysis would apply, for example, to someone who has a special responsibility towards the people who get lost, such as camp counselors at a summer camp for children, or a babysitter in charge of an outing. Someone like that would have a duty to use reasonable care to keep the children safe, and would be judged to determine if they met that standard in light of all of the facts and circumstances.
This is something that can't really be prejudged for any particular set of facts. Two juries hearing exactly the same evidence could come to different conclusions on this question and either outcome could be upheld on appeal. The inquiry is too dependent on a rich set of overall facts for the information in the question to be outcome determinative.