The question is this I've been trying to get my head around this but simply cannot and am hoping you might get me going.
Q: The Store is open from $8$ am-$8$ pm every single day. $X$ represents the hours, $f(x)$ is how many customers there are.
What time can you mathematically presume the most customers arrived and how many customers are there?
Hint: Since $f(x)$ denotes the number of customers, in order to find the most customers, you need to calculate the maximum value that $f$ takes within its domain (from 8 in the morning until 8 at night). Then, the particular $x_0$ for which $f(x_0)$ is maximum, will be the time when most customers arrived.
$$f(x)=-4x^2+111.26x-625.16$$ The commas threw me way off. Perhaps it's different in other countries, but when you want to say $\frac12$ as a decimal, for example, it will be $0.5$. That's just a tip for later. For example: $14.342,343$ is $14,342.343$ here in the U.S. and in most places around the world.
Anyhow, for this problem, what we want to find is the maximum of $f(x)$ in that interval. Perhaps the first thing we should look for is absolute maximums.
Absolute maximums are when the slope is equal to $0$, and when the slope of that slope is decreasing (goes from positive to negative). In other words, $f'(x)=0$ and $f''(x) < 0$. Or you can just check that $f'(x)$ goes from positive to negative.
$$f'(x)=-8x+111.26$$ $$f''(x)=-8$$
I'm assuming you know how to differentiate... It seems that the slope is always decreasing, so whatever we find for $f'(x)=0$ is going to be an absolute maximum. Let's solve it then:
$$0=-8x+111.26$$ $$x=13.9075$$
That's our answer. Anything to the right or left of it will be lower (because the second derivative is negative). That's our time: $13.9075=13:54:27\approx 1:54\text{ P.M.}$ That's another thing, in America we don't use a 24-hour clock. We denote morning as A.M. and afternoon as P.M. For example: 14:53 is 2:53 P.M.
