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I would like to ask the way how we consider the range of values of x for what the function is decreasing in a cubic equation by differentiation.

The question is to find the range of values of x for what the function is decreasing. $$P(x)=2x^3-81x^2+840x$$

The differentiated value is $6x^2-162x+840$. And the answers of x for what the function is decreasing are $x<20$ and $x<7$ but why do we have to consider the range is $7<x<20$ .

Kyooo
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  • The function is decreasing where the derivative is negative. The derivative is NOT negative for every $x < 20$, and it is positive, not negative, for $x < 7$. Where did you get the idea that once you found the two roots $7$ and $20$, the next step was just to put $x <$ in front of them without paying any attention to what the derivative is actually doing? – Paul Sinclair Jun 14 '23 at 17:03
  • But when I tried to draw a graph, the function is decreasing between the values of x, 7 and 20. – Kyooo Jun 14 '23 at 22:13
  • And you don't see that what I said is exactly in line with that? To find where a differentiable function is decreasing, you find where its derivative is negative. To do that, you note that the derivative cannot switch from positive to negative or vice versa without passing through $0$, so you find where the derivative is $0$. Then between and outside those locations you test whether the derivative is positive or negative a random point in each. It will have the same sign over the whole interval. – Paul Sinclair Jun 14 '23 at 22:46
  • So the key point is to know the two points and it does not matter whether the sign is greater or less than one. Arrr I think I get it. So if the derivative is less than zero, we will get the points where the function is stationary but the points will not mean that the function is decreasing at those values of x. And we will know the nature by calculating the second derivative. Am I stating that right? – Kyooo Jun 15 '23 at 10:28
  • Huh? Where do you see any of that? It directly contradicts what I said. It appears you are grabbing terminology out of your book and flinging it around, without paying any more attention to what the book is actually saying than you've paid to what I actually said, as those terms are related to finding maxima and minima, not where the function is increasing or decreasing. – Paul Sinclair Jun 15 '23 at 11:31
  • Once again: find where the derivative is $0$. Those points divide the real line into intervals. On each of those intervals, the derivative is either positive throughout or negative throughout. You can figure out which is which simply by evaluating the derivative at one point in each of the intervals. The intervals where the derivative is positive are where the function is increasing. The intervals where the derivative is negative are where the function is decreasing. – Paul Sinclair Jun 15 '23 at 11:33

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