A point of inflection occurs when #f''(x)# switches from positive to negative, or vice versa. This correlates to #f(x)# switching concavity.
Possible points of inflection occur when #f''(x)=0#. So, to find the possible points of inflection, differentiate #f(x)# twice.
#f(x)=x^4#
#f'(x)=4x^3#
#f''(x)=12x^2#
Set #f''(x)=0# to find possible points of inflection.
#12x^2=0#
#x=0#
Create a sign chart to determine if the second derivative switches signs when #x=0#.
#color(white)(-----------)0#
#f''(x)color(white)(---)larr----------rarr#
#color(white)(------)"POSITIVE"color(white)(---)"POSITIVE"#
#f''(x)# doesn't change signs at #x=0#, so there's not a point of inflection when #x=0#. Since #x=0# was the only possible point, we know that #x^4# has no points of inflection.
graph{x^4 [-10.04, 9.96, -2.2, 7.8]}
