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

1 Answer
Feb 7, 2017

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.