# Using the algorithm for curve sketching i.e first derivative and second derivatives.  Explain what the graph of y=x^(5/3)-2x^(2/3) would look like. My teacher said there would be a cusp in the graph.  How would I know that?

Usually you have a cusp, when

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i) Let's start by finding the inflection points.

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Point of inflection

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I will use the unsimplified y' to find y''

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since the denominator is always non negative we only need to study the numerator.

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x=4/5 falls in the firt interval, thus we know that we have a relative minimum.

ii) For the Cusp.

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we notice that x=0 will make the denominator zero, thus y' undefined. let's study the limit at this point.

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because the numerator will be negative, as well as the denominator.

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because the numerator will be negative but denominator positive

Hence we have a cusp at x=0, in other words we have a vertical tangant.

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