# How do you sketch the graph by determining all relative max and min, inflection points, finding intervals of increasing, decreasing and any asymptotes given #f(x)=x-x^(2/3)(5/2-x)#?

##### 1 Answer

Dec 18, 2016

See explanation and graph.

#### Explanation:

Rearranging,

The graph passes through the origin (0, 0)

x-intercept ( y = 0 ): 1.4, nearly.

As

So, there are no asymptotes.

At x =0, y' has infinite discontinuity. It changes from

and becomes 0 near x = 0.5, for sign change, from

For

again .

There is a turning point near x = 0.5, for

the local minimum

,Relative maximum y = 0, at the cusp (0, 0).

The origin is a cusp and the tangent does not cross the curve,

there is no point of inflexion

graph{x^(2/3)(-2.5+x^(1/3)+x) [-2.5, 2.5, -1.25, 1.25]}