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I'm trying to find the roots for this function:

$$ f'(x) = \dfrac{w_0w_1y_0y_1\left(\frac{x_0}{x+x_0}\right)^\frac{w_0}{v_0}\left(\frac{x_1}{y_0\left(1-\left(\frac{x_0}{x+x_0}\right)^\frac{w_0}{v_0}\right)+x_1}\right)^\frac{w_1}{v_1}}{v_0v_1\left(x+x_0\right)\left(y_0\left(1-\left(\frac{x_0}{x+x_0}\right)^\frac{w_0}{v_0}\right)+x_1\right)}-1 $$

Which is the first derivative of this function of which I'm trying to find the maximum:

$$ f(x) = y_1\left(1-\left(\dfrac{x_1}{y_0\left(1-\left(\frac{x_0}{x+x_0}\right)^\frac{w_0}{v_0}\right)+x_1}\right)^\frac{w_1}{v_1}\right)-x $$

I'm simply trying to use an online derivative calculator (https://www.derivative-calculator.net/) but it says no roots can be found for the derivative, which is strange since there is clearly a local/global maximum when plotted.

To see the plot in the website I've linked follow this steps:

  1. Copy-paste this formula on the box

y1*(1-(x1/(x1+(y0*(1-(x0/(x0+x))^(w0/v0)))))^(w1/v1))-x

  1. Click on Go!
  2. At the bottom on the page you can see the graph, you can use this example parameters:

x0 = 800

y0 = 500,000

x1 = 600,000

y1 = 1417

w0 = 0.8

v0 = 0.2

w1 = 0.15

v1 = 0.85

  1. Now you can see that there is a global maximum on $ f(3.717) = 0.0772$

Does this function really have no roots?

Thanks

Hiperfly
  • 101
  • I don't really have time at all to look into this in any detail, but I will say that it's possible to have a global maximum but no zero. Consider $f(x)=\frac{\sin(x)}{x} + 1, f(0) = 2$ (with the extra condition to make it continuous.) Or $e^{-x^2}$. Existence of a global maximum $\nRightarrow$ existence of a root. – Moni145 Jun 11 '21 at 09:29
  • maximums occur at critical points, as above, so anywhere you have the derivative 0 or undefined. Thus you have an automatic critical point at $-x_0$ and maybe some more, too tired to look further – Alan Jun 11 '21 at 09:51
  • I really appreciate both your inputs. In this case $x > 0$ always, so function is defined and is continuous for all the acceptable values of $x$. Also $w0,v0,w1,v1$ are always positive (non-zero). – Hiperfly Jun 11 '21 at 10:02
  • By the way, is not possible at all then to calculate some sort of roots for this function, or something similar? – Hiperfly Jun 11 '21 at 10:10

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