Maximum and minimum problems

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.

• In itself, the given polynomial doesn't have global maximum or minimum value, as you have seen yourself. But looking at the problem definition, surely the "number of hours" cannot be negative, so we first of all restrict ourselves to $x\geq 0$. Then we get a minimum value but not a maximum. I think there should be another condition for the maximum imaginable number of hours, in order to restrict the value of the polynomial. – Matti P. Mar 16 '18 at 6:26
• You must be missing a condition $\,x \le 10\,$ somewhere. – dxiv Mar 16 '18 at 6:27
• @dxiv Good point. Very clever! – CiaPan Mar 16 '18 at 6:53

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.