I see that when $f'(x)=0$ we have the value of $x$ where the value of $y$ is the largest or smallest. But in this task, what is the meaning of the lowest value of $f'(x)$? I don't understand how we can take the $x$ value from the derivative and the $y$ value from the function and get this information, and especially in this question I don't even know what the information is supposed to mean.
The derivative of a function $f$ describes the gradient of the function at that point. When $f'(x) = 0$, this corresponds to when the curve has zero gradient, or when the tangent line to $f$ at $x$ is horizontal. Note that this does not correspond with the largest or the smallest values of $y$ in general; for example, the function $f(x) = x^{3}$ has derivative $f'(x) = 3x^{2}$ so that $f$ has zero gradient at $x = 0$, but this has $y$-coordinate $y = 0$ so it is neither the largest nor the smallest value of $y$ on $f$ if $f$ is defined on the whole of the real line.
Note that a positive $f'(x)$ value corresponds to $f$ increasing in $y$ as $x$ increases, and a negative $f'(x)$ value corresponds to $f$ decreasing in $y$ as $x$ increases. Thus, when $f'(x)$ is a large negative number, this means that $f$ is decreasing quickly, so it will be going down steeply (as $x$ increases). Similarly, when $f'(x)$ has a large positive number, this means that $f$ is increasing quickly, so it will be going up steeply (as $x$ increases).
In terms of your example, a large positive $f'(x)$ value corresponds to a large rate of change of production, so rate of production is increasing as the days pass. A large negative $f'(x)$ value also corresponds to a large rate of change of production but in a negative way, so rate of production is decreasing as the days pass. If $f'(x)$ is zero, then the rate of production stays the same.