# 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...

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?

*print*Print*list*Cite

### 1 Answer

Usually you have a cusp, when

or

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

``

so ``

``

Point of inflection

``

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

``

``

``

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

``

``

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

ii) For the Cusp.

``

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

``

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

``

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.