What is the point of maximum growth rate for the logistic function f(x) ?

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1 Answer
Mar 6, 2018

#x(6.93147, 7.5)#

Fourth answer down is correct.

Explanation:

.

#y=15/(1+4e^(-0.2x))# whose graph is:

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We know that in the logistic function in the form of:

#y=C/(1+Ae^(-Bx)#

#C# is the maximum value of #y#, the graph of the function is symmetric, and the point of maximum growth occurs at halfway between #0# and the maximum #y#, i.e. at #y=C/2# which is the #y#-coordinate of that point.. Therefore, to find the #x#-coordinate of it, we plug in #C/2# for #y# and solve for #x#:

#15/2=15/(1+4e^(-0.2x))#

The numerators of both sides are equal. Therefore, the denominators must be equal:

#2=1+4e^(-0.2x)#

#e^(-0.2x)=1/4#

#lne^(-0.2x)=ln(1/4)#

#(-0.2x)lne=ln(1/4)#

#(-0.2x)(1)=ln(1/4)#

#-0.2x=ln(1/4)#

#x=ln(1/4)/-0.2=(-1.386294)/-0.2=6.93147#