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For example, for the function $f(x) = x^{\frac{4}{3}}-4x^{\frac{1}{3}},$ there is a vertical tangent at $(0, 0),$ but it is also here the the concavity changes from negative to positive, as you can see from the second derivative. Is this point both a point of inflection and a vertical tangent?

My teacher's notes only state that it is a vertical tangent, which is confusing because I would have expected him to state that it is also a POI. Does being a vertical tangent imply that there is a POI?

Jack Pan
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  • What exactly do you mean by "vertical tangent"? the function isn't differentiable at $;x=0;$ ... – DonAntonio Mar 13 '17 at 18:48
  • When your function has a vertical tangent it can also have a point of inflection or a cusp. As @DonAntonio notes your function is not differentiable in that point so take care when you need to express the equation of the tangent line. – N74 Mar 14 '17 at 13:55

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