# Determine the point of inflection of p(t) = 20/(1 + 3e^-0.02t).

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The point of inflection is when a curve changes concavity. This equates to the second derivative being zero at points of inflections.

We can use the chaing rule that `(d)/(dx)1/u = -1/u^2(du)/(dx)`

`(dp(t))/(dt)=20(-1/(1+3e^(-0.02t))^2)(d)/(dt)(1+3e^(-0.02t))`

` = -20/(1+3e^(-0.02t))^2(3(-0.02)e^(-0.02t))=(1.2e^(-0.02t))/(1+3e^(-0.02t))^2`

`(d^2(p(t)))/(dt^2)=(1.2(-0.02)e^(-0.02t))/(1+3e^(-0.02t))^2+`

`(1.2e^(-0.02t))*(-2/(1+3e^(-0.02t))^3)*(3(-0.02)e^(-0.02t))`

`=(0.024e^(-0.02t))/(1+3e^(-0.02t))^3(-1+3e^(-0.02t))`

This can be zero when `(-1+3e^(-0.02t)) = 0`

We get `3e^(-0.02t)=1`

`e^(-0.02t)=1/3`

`-0.02t=ln(1/3)=-ln(3)`

`t = (-ln(3))/(-0.02)=50ln(3)~~54.93` This is the point of inflection.