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The function f is continuous on the interval (0,16), and f is twice differentiable except at x=5 where the derivatives are undefined. Information about the first and second derivatives of f for values of x in the interval (0,16) is given in the table above. At what values of x in the interval (0,16) does the graph of f have a point of inflection?.

Respuesta :

Using it's definition, it is found that the function f(x) has a point of inflection at:

A. x = 8 only.

What are the points of inflection of a function?

  • The critical points of a function are the values of x for which:

[tex]f^{\prime\prime}(x) = 0[/tex]

  • Additionally, there has to be a change in the sign of [tex]f^{\prime\prime}(x)[/tex]

Researching the problem on the internet, it is found that:

  • For 0 < x < 5, [tex]f^{\prime\prime}(x) > 0[/tex].
  • For x = 5, [tex]f^{\prime\prime}(x)[/tex] is undefined.
  • For 5 < x < 8, [tex]f^{\prime\prime}(x) < 0[/tex].
  • For x = 8, [tex]f^{\prime\prime}(x) = 0[/tex].
  • For 8 < x < 12, [tex]f^{\prime\prime}(x) > 0[/tex].
  • For x = 12, [tex]f^{\prime\prime}(x) = 0[/tex].
  • For 12 < x < 16, [tex]f^{\prime\prime}(x) > 0[/tex].

The two conditions, [tex]f^{\prime\prime}(x) = 0[/tex] and a change in the signal of [tex]f^{\prime\prime}(x)[/tex] are only respected at x = 8, which is the lone inflection point.

You can learn more about points of inflection at https://brainly.com/question/10352137

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