Critical Points of Inflection

Key Questions

  • Do points of inflection have to be differentiable?

    That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:

    enter image source here

    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.

    Wataru · 5 · Sep 20 2014
  • Can points of inflection be extrema?

    That is a good question! I had to revisit the definition in the Calculus book by Stewart, which states:

    enter image source here

    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.

    Wataru · 1 · Sep 19 2014

Questions

Graphing with the Second Derivative