# 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)?

Dec 18, 2016

See explanation and graph.

#### Explanation:

Rearranging,

$y = f \left(x\right) = \left({x}^{\frac{5}{3}} + x - 2.5 {x}^{\frac{2}{3}}\right)$

The graph passes through the origin (0, 0)

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

As $x \to \pm \infty , y \to \pm \infty$.

So, there are no asymptotes.

$y ' = \frac{5}{3} {x}^{\frac{2}{3}} + 1 - \frac{\frac{5}{3}}{x} ^ \left(\frac{1}{3}\right)$

At x =0, y' has infinite discontinuity. It changes from $\infty$ to$- \infty$.

and becomes 0 near x = 0.5, for sign change, from $-$ to +

For $x < 0. y \uparrow , x \in \left(0 , 0.5\right) , y \downarrow , \mathmr{and} x > 0 , 5$ ( nearly), $y \uparrow$,

again .

There is a turning point near x = 0.5, for

the local minimum $= y \left(0.5\right) = - 0.76$, nearly.,

,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]}