Question

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

Answer 1

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

Explanation:

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))`