Determining Points of Inflection for a Function
Key Questions

#f(x)=x^3+x# By taking derivatives,
#f'(x)=3x^2+1# #f''(x)=6x=0 Rightarrow x=0# ,which is the
#x# coordinate of a possible inflection point. (We still need to verify that#f# changes its concavity there.)Use
#x=0# to split#(infty,\infty)# into#(infty,0)# and#(0,infty)# .Let us check the signs of
#f''# at sample points#x=1# and#x=1# for the intervals, respectively.
(You may use any number on those intervals as sample points.)#f''(1)=6<0 Rightarrow f# is concave downward on#(infty,0)# #f''(1)=6>0 Rightarrow f# is concave upward on#(0,infty)# Since the above indicates that
#f# changes its concavity at#x=0# ,#(0,f(0))=(0,0)# is an inflection point of#f# .I hope that this was helpful.

No. Consider
#f(x)=x#  this function's concavity does not change throughout the entire run of the function.All polynomials with odd degree of 3 or higher have points of inflection, and some polynomials of even degree (again, higher than 3) have them. The best way to determine if a function has a point of inflection is to look at its second derivative  if the second derivative can equal zero, the original function has a point of inflection.
Questions
Graphing with the Second Derivative

Relationship between First and Second Derivatives of a Function

Analyzing Concavity of a Function

Notation for the Second Derivative

Determining Points of Inflection for a Function

First Derivative Test vs Second Derivative Test for Local Extrema

The special case of x⁴

Critical Points of Inflection

Application of the Second Derivative (Acceleration)

Examples of Curve Sketching