Determining Points of Inflection for a Function

Key Questions

  • How do you find the inflection points for the function f(x)=x^3+x?

    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.

    Wataru · · Oct 10 2014
  • Do all functions have points of inflection?

    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.

    CJ · 1 · Oct 12 2014

Questions

Graphing with the Second Derivative