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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.

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Hills
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    Do you understand what the derivative (also called the gradient) means geometrically? – Bilbottom May 02 '18 at 10:27
  • I understand that the derivative is the slope of the tangent in one point on f. And that if $f'(x)=0$ the slope is zero and then the tangent is horizontal, and therefor this must be a bottom or top point? – Hills May 02 '18 at 10:31
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    Yes, the tangent is horizontal when the gradient is zero, but note that does not correspond with the largest or the smallest values of $y$ in general. However, with this intuition, can you see why the smallest $f'(x)$ value corresponds to when $x = 12.5$? – Bilbottom May 02 '18 at 10:33
  • @BillWallis So the tangents biggest slope is in this point? That means the production after this point is reduced the most onward? – Hills May 02 '18 at 10:38
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    Yes, the tangents largest negative slope is at this point, so at this point the rate of production reaches it's lowest point. Note that the rate of production does start to increase again after this point. – Bilbottom May 02 '18 at 10:40
  • Ah, thank you very much! – Hills May 02 '18 at 10:41

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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.

Bilbottom
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