Question

How do you sketch the curve `y=x^5-x` by finding local maximum, minimum, inflection points, asymptotes, and intercepts?

Answer 1

See explanation and Socratic graph.

Explanation:

Sufficient data for studying the shape and extent of the graph:

x-intercepts `( y = 0) : 0, +-1`

y-intercept ( x = 0 ) : 0.

`y'=5x^4-1 = 0`, at `x = +-1/sqrt (sqrt5)=+-0.66874`, 4early

`y''= 20x^3 =0`, at x = 0.

The turning points are `(+-1/sqrtsqrt5, +-(4/5)/(sqrtsqrt5))`, giving

local ( y'' < 0 ) maximum `(4/5)/sqrtsqrt5=0.535` and

local ( y'' > 0 ) minimum -(4/5)/sqrtsqrt5=-0.535#, nearly.

See the first graph for these features.

At x = 0, y'''=y'''=0 and y'''''= 120 `ne 0`.

See the tangent crossing the curve at POI, in the not-to-scale

second ad hoc ( for the purpose ) graph.

As `to +-oo, y = x^5(1-1/x^4) to +-oo`.

graph{(x^5-x-y)((x-.67)^2+(y+.535)^2-.002)((x+.67)^2+(y-.535)^2-.002)=0 [-5, 5, -2.5, 2.5]}

graph{(x^5-x-y)(y+x)=0 [-1, 1, -2.5, 2.5]}

Tangent at the POI (0, 0) crosses the curve.