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:

  

  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:

  

  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

• Can a point of inflection be undefined?
• Can points of inflection be extrema?
• Do points of inflection have to be differentiable?
• What do points of inflection represent on a graph?
• Find the points of inflection of the curve `y=(1+x)/(1+x^2)`?
• How do you find the points of inflection of the curve `y=e^(x^2)`?
• How do you find the point of inflection of a cubic function?
• What is difference between critical points and inflection points?
• How do you locate the critical points of the function `f(x) = x^3 - 15x^2 + 4` and use the Second Derivative Test to determine whether they correspond to local maxima, local minima, or neither?
• Are the inflection points where f'(x) = zero or where the graph changes from concave up to concave down?
• How do you determine intervals on which the function is concave up or down and find the points of inflection for `y=(x^2-7)e^x`?
• How do you find all points of inflection given `y=x^3-2x^2+1`?
• How do you find all points of inflection given `y=x^2/(2x+2)`?
• How do you find all points of inflection given `y=x^4-3x^2`?
• How do you find all points of inflection given `y=-x^5+2x^3+4`?
• How do you find all points of inflection given `y=x^5-2x^3`?
• How do you find all points of inflection given `y=-(2x)/(x+3)`?
• How do you find all points of inflection given `y=-x^4+3x^2-4`?
• How do you find all points of inflection given `y=-x^3/(x^2-1)`?
• How do you find all points of inflection given `y=-2sinx`?
• How do you find all points of inflection given `y=-(x+2)^(2/3)`?
• How do you find all points of inflection given `y=-(4x+20)^(1/3)`?
• How do you find all points of inflection given `y=x^3+9x^2+24x+22`?
• How do you find all points of inflection given `y=((x-3)/(x+1))^2`?
• How do you find all points of inflection given `y=2x^2+4x+4`?
• How do you find all points of inflection given `y=-sin(2x)`?
• How do you determine the intervals where `f(x)=3x^2-x^3/3` is concave up or down?
• How do you determine the intervals where `f(x)=3x-4` is concave up or down?
• How do you determine the intervals where `f(x)=x^2-4x+7` is concave up or down?
• How do you determine the intervals where `f(x)=x^4-x^2` is concave up or down?
• How do you determine the intervals where `f(x)=x^(2/3)+3` is concave up or down?
• How do you find all critical point and determine the min, max and inflection given `f(x)=x^2-8x-10`?
• How do you find all critical point and determine the min, max and inflection given `D(r)=-r^2-2r+8`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=3x^2-4x+1`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=2x^3-x^2+1`?
• How do you find all critical point and determine the min, max and inflection given `V(w)=w^5-28`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=x^3+x^2-x`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=x^3-6x^2+9x+8`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=x^4-4x^3+20`?
• How do you find all critical point and determine the min, max and inflection given `f(x)=x^4`?