Critical Points of Inflection
Key Questions

That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:
My answer to your question is no, a function does not need to be differentiable at a point of inflection; for example, the piecewise defined function
#f(x)={(x^2,if x<0), (sqrt{x},if x ge0):}# is concave upward on
#(infty,0)# and concave downward on#(0,infty)# and is continuous at#x=0# , so#(0,0)# is an inflection point but not differentiable there. 
That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:
My answer to your question is yes, an inflection point could be an extremum; for example, the piecewise defined function
#f(x)={(x^2,if x<0), (sqrt{x},if x ge0):}# is concave upward on
#(infty,0)# and concave downward on#(0,infty)# and is continuous at#x=0# , so#(0,0)# is an inflection point and a local (also global) minimum.
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