Question

How do you find the inflection points of the graph of the function: `f(x) = (6x)/(x^2 + 16)`?

Answer 1

`x = 0, f(0) = 0 " and " x = pm 4sqrt(3), f(pm 4sqrt(3)) = pm (3sqrt(3))/8`

Explanation:

A point of inflection can be found when the second derivative of f(x) is equal to zero i.e. `f''(x) = 0`

Using quotient rule:
`(df)/dx = (6(x^2+16)- 12x^2)/(x^2+16)^2 = -(6(x^2-16))/(x^2+16)^2`
`(d^2f)/dx^2 = -(12x(x^2+16)-24x(x^2-16))/(x^2+16)^3 = (12x(x^2-48))/(x^2+16)^3 = 0`

`x = 0, f(0) = 0 " and " x = pm 4sqrt(3), f(pm 4sqrt(3)) = pm (3sqrt(3))/8`