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Thus, the engagement score after 6 minutes is 19.
\boxed{19}Question: A meteorologist models the change in atmospheric pressure over a 24-hour period with the function $ P(t) = t^3 - 6t^2 + 9t + 10 $, where $ t $ represents time in hours after midnight. At what time(s) does the pressure reach a local minimum or maximum?
Solution: To find the local extrema of the function $ P(t) = t^3 - 6t^2 + 9t + 10 $, we first compute its derivative:
P'(t) = 3t^2 - 12t + 9.
Set $ P'(t) = 0 $ to find critical points:
3t^2 - 12t + 9 = 0 \Rightarrow t^2 - 4t + 3 = 0.
Factoring gives:
To determine the nature of these critical points, we use the second derivative:
P''(t) = 6t - 12.
Evaluate at the critical points: