What are the points of inflection of #f(x)=x^5 - 2x^3 +4x#?

1 Answer
Apr 21, 2017

The points of inflection of #f(x)# are at
#x=-sqrt(3/10), x=0,# and #x=sqrt(3/10)#

Explanation:

Use power rule to find #f''(x)#

#f(x)=x^5-2x^3+4x#
#f'(x)=5x^4-6x^2+4#
#f''(x)=20x^3-6x#

For points of inflection, set second derivative equal to zero:
#0=f''(x)#
#0=2(x)(10x^2-3)#

Using zero product rule :
#color(blue)(x=0#

#10x^2-3=0#
#x^2=3/10#
#color(blue)(x=+-sqrt(3/10)#

Double check that these answers are actual points of inflection by testing values in between these intervals (or draw a "sign line", as some call it).

Thus, the points of inflection of #f(x)# are at
#x=-sqrt(3/10), x=0,# and #x=sqrt(3/10)#