Basic Examples 
(4)
 

Find the inflection points of a cubic function:
In[33]:=
func=x^3;​​inflPts=
[◼]
InflectionPoints
[func,x]
Out[34]=
{{x0}}
Plot the function and its inflection points found above:
In[35]:=
Plot[{func},{x,-3,3},EpilogJoin[{Red,PointSize@Large},Point[{#〚1,2〛,func/.#}]&/@inflPts]]
Out[35]=
-3
-2
-1
1
2
3
-20
-10
10
20
Repeat the calculation, classifying inflection points:
In[36]:=
[◼]
InflectionPoints
[func,x,"Properties"]
Out[36]=
{{x0,Properties{Stationary,Rising}}}
———
Find and classify the inflection points of a polynomial function:
In[55]:=
func=1+3
2
x
+
3
x
;​​inflPts=
[◼]
InflectionPoints
[func,x,"Properties"]
Out[56]=
{{x-1,Properties{Non-stationary,Falling}}}
Plot the function and its inflection points:
In[57]:=
Plot[{func},{x,-4,2},EpilogJoin[{Red,PointSize@Large},Point[{#〚2〛,func/.#}]&/@inflPts〚All,1〛]]
Out[57]=
-4
-3
-2
-1
1
2
-5
5
10
———
Find and classify the inflection points of another polynomial function:
In[37]:=
func=(x-1)(x-2)(x-3)(x-3.2);​​inflPts=
[◼]
InflectionPoints
[func,x,"Properties"]
Out[38]=
{{x1.79338,Properties{Non-stationary,Rising}},{x2.80662,Properties{Non-stationary,Falling}}}
Plot the function and its inflection points:
In[39]:=
Plot[{func},{x,0.7,3.9},EpilogJoin[{Red,PointSize@Large},Point[{#〚2〛,func/.#}]&/@inflPts〚All,1〛]]
Out[39]=
1.0
1.5
2.0
2.5
3.0
3.5
-0.5
0.5
1.0
1.5
2.0
———
Find and classify the inflection points of a trigonometric function:
In[42]:=
func=Sin[x];​​inflPts=
[◼]
InflectionPoints
[func,x,"Properties"]
Out[43]=
x
2π

1
if

1
∈
,Properties{Non-stationary,Rising},x
π+2π

1
if

1
∈
,Properties{Non-stationary,Falling}
Plot the function and a single cycle's worth of its inflection points:
In[44]:=
Plot[{func},{x,-5,7},EpilogJoin[{Red,PointSize@Large},Point[{#〚2〛,func/.#}]/.

1
0&/@inflPts〚All,1〛]]
Out[44]=
-4
-2
2
4
6
-1.0
-0.5
0.5
1.0

Options 
(2)
 


Properties and Relations 
(3)
 
