How do you find the inflection points of the graph of the function: #f(x) = ((4 x)/e^(9 x))#?

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Answer 1 · 2015-07-29T01:26:00

Inflection points are points on the graph at which the concavity changes. So investigate concavity.

To investigate concavity, we will look at the sign of the second derivative.

#f(x) = (4x)/e^(9x)#

#f'(x) = (4e^(9x) - 4xe^(9x)*9)/e^(18x)#

# = (4-36x)/e^(9x)#

#f''(x) = ((-36)e^(9x) - (4-36x)e^(9x)*9)/e^(18x)#

# = (36(9x-2))/e^(9x)#

The factors #36# and #e^(9x)# are always positive, so the sign of #f''# is the same as the sign of #9x-2# which changes at #x=2/9#

The point #(2/9, f(2/9))# is the only inflection point.

Since #f(2/9) = 8/(9e^2)#,

the inflection point is: #(2/9, 8/(9e^2))#