Can a function have both a vertical tangent and cusp? Does The function $3x^{1/3}(x+2)$ have a vertical tangent and if so why? I believe that it has a cusp.
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Typically terminology distinguishes between cusps and vertical tangents. A cusp is where slopes approach $+\infty$ on one side, and $-\infty$ on the other. For vertical tangents, slopes approach $+\infty$ on both sides, or $-\infty$ on both sides.
For example, the real cube root function has a vertical tangent (with slopes approaching $+\infty$) at $(0,0)$, whereas $x\mapsto \sqrt[3]{x^2}$ has a cusp (slopes on the left approaching $-\infty$, slopes on the right approaching $+\infty$) at $(0,0)$.
Your function behaves like the real cube root function at $(0,0)$, and has a vertical tangent there, not a cusp. That is, unless by $x^{1/3}$ you meant something other than the real cube root function.
Jonas Meyer
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$y'(x) = \dfrac{x+2}{\sqrt[3]{x^2}} + 3\sqrt[3]x$, and $y'(0) = \text{undefined}$ .Thus there is a vertical tangent at $(0,0)$.
DeepSea
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So a vertical tangent is when y' is undefined? – Nov 19 '14 at 04:50
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according to my book the vertical tangent occurs when "slopes of secant lines approach positive/negative infinity from both sides" (Finney Demana Waits Kennedy). The function $3x^{1/3}(x+2)$ doesn't satisfy said condition. – Nov 19 '14 at 04:54
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@AmandaKelius, I agree with that book's definition. As the function is defined at zero it can't have a vertical tangent (which I'd rather call asympote as Mahidevran does, though perhaps these two are different). – Timbuc Nov 19 '14 at 05:01
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@Timbuc: That is incorrect. $\sqrt[3]{x}$, if defined on $\mathbb R$ to be the inverse of $x^3$, does have a vertical tangent at $(0,0)$. Same is true if you multiply it by $3(x+2)$. – Jonas Meyer Nov 19 '14 at 05:37
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@Amanda: Why are you claiming it doesn't satisfy the condition? – Jonas Meyer Nov 19 '14 at 05:42
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@JonasMeyer, I can't really say. First, that "vertical tangent" thing sounds a little weird to me. I'd go for asymptote but I realize these may be two different things, though I can't really think ofanything really important or interesting about "verticl tangents". Perhaps they have some use similar to asymptotes, but I don't know it. – Timbuc Nov 19 '14 at 13:05
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@Timbuc: Tangent lines to a curve and asymptotes are very different. – Jonas Meyer Nov 19 '14 at 13:32
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Oh, I know that now, @JonasMeyer...yet I can't see how tangent lines can help for something, either invertigating the function, to draw it or whatever. – Timbuc Nov 19 '14 at 13:37
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OK. I suppose this isn't the place to convince you that tangents are helpful, so I'll stop. – Jonas Meyer Nov 19 '14 at 13:40