How do you sketch the curve #y=x^5-x# by finding local maximum, minimum, inflection points, asymptotes, and intercepts?
1 Answer
See explanation and Socratic graph.
Explanation:
Sufficient data for studying the shape and extent of the graph:
x-intercepts
y-intercept ( x = 0 ) : 0.
The turning points are
local ( y'' < 0 ) maximum
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
See the tangent crossing the curve at POI, in the not-to-scale
second ad hoc ( for the purpose ) graph.
As
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.