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

Jan 12, 2017

$x = - 0.544120341 \mathmr{and} 1.72049868$, nearly

#### Explanation:

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

$x = - 0.6 \mathmr{and} 1.75$

Using these as starters for Newton-Raphson iteration,

the two POI are $x = - 0.544120341 \mathmr{and} 1.72049868$, 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]}