What are the points of inflection, if any, of #f(x) = x^5/20 - 5x^3 + 5 #?

1 Answer
Apr 20, 2018

The points of inflection are #x= -sqrt30, 0, and sqrt(30)#.

Explanation:

#f(x) = (x^5/20) - 5x^3 + 5#

Points of inflection are where the second derivative of a function is equal to 0, and where the second derivative of a function switches signs. So first you find the second derivative and set it equal to 0.

# f'(x) = (5x^4/20) - 15x^2 = (x^4/4) -15x^2 #
#f''(x) = (4x^3/4) - 30x = x^3 - 30x #
#x^3 - 30x = 0#

Factor out the x.
#x(x^2 - 30) = 0#
so #x=0# and #x = +- sqrt(30) #

To check if these are all inflection points, you check if the signs change around them. I plugged in #x=-6, -1, 1#, and #6#, but you can do this with any points outside and between the three answers. The signs changed around all 3 points, so all 3 are points of inflection.