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

1 Answer
Aug 11, 2015

# 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 #