What are the points of inflection of #f(x)=x^6 + 3x^5 - x^4 - 40x^3 - 60x^2 + 8x + 5 #?

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Answer 1 · 2017-01-12T16:24:24

#x =-0.544120341 and 1.72049868#, nearly

The first graph is for f and the second is for f''.

The second pinpoints zeros of f'', for points of inflexion (POI)

It is evident that the tangent crosses the curve near

#x = -0.6 and 1.75#

Using these as starters for Newton-Raphson iteration,

the two POI are #x =-0.544120341 and 1.72049868#, nearly
.graph{x^6+3x^5-x^4-40x^3-60x^2 [-50, 50, -2000, 2000]}

graph{30x^4+60x^3-12x^2-240x-120 [-2.5, 2.5, -1.25, 1.25]}