The number of people, $P$, visiting a certain beach on a particular day in January depends on the number of hours, $x$, that the temperature is below thirty degrees, according to the rule $P=x^3-12x^2+21x+105$ where $x \geq 0$. Find the value of $x$ for the maximum and minimum number of people who visit the beach.
In answering this question, I found the derivative of the rule, made it equal to $0$, and solved for $x$ ($x=1$ and $x=7$). I then found that $(1, 116)$ is the maximum and $(7,8)$ is the minimum. However, I am confused why these local max and minimums are used to answer the question. The graph shows that as $x$ gets larger, $P$ becomes greater than $116$ (shown on graph), meaning $x = 1$ is not the maximum point of the graph? I am unsure as to why $x = 1$ is then defined as the maximum.
I agree with you that $x=1$ is not the solution.
$x$ carries a physical meaning here, the number of hours in a day the temperature is below $30$ degrees. Hence I would think that $0 \leq x \le 24$ and I would answer $x=24$ is optimal unless there are some other constraints.