Question

What are the points of inflection, if any, of #f(x)=x^3 - 12x^2 #?

Answer 1

`The` point of inflexion (POI) is ( 4, -`128 ). I could not drag the graph to `y < -90`, tosee the POI that is far down below.

Explanation:

`y=x^3-12x^2=x^2(x-12)=0`, for x = 0 and x = 12.

`y'=3x^2-24x=3x(x-8)=0`, for x = 0 and x = 8.

These are turning points. The first point is (0, 0). The second is far

down at `(8, -256)`, to rise for `y to oo, as x to oo`, through (12, 0)..

#y''=6x-24=0, for x = 4. This point ( 4, -128) is a candidate for point of

inflexion (POI).

#y'''= 6. y''' not 0 when y'' = 0 is a sufficient condition,

for this point (4, -128) to be a POI.

Of course, as `x to -oo, y to -oo`.

Using contracting scale 1 for 20 , exclusively for y and 1 for 2 in x,

this POI and other features that are not revealed by this graph,

could be brought lo light, from hiding.

In this edition, I have managed contraction in another graph ( the

first below ) to reveal more features. Yet, for the the POI (point of

inllexion ) location, local zooming has to be made at `(4, -128)`.

graph{y=x^3-12x^2 [-324.5, 324.5, -166.8, 157.7]}
graph{y=x^3-12x^2 [-9.915, 10.085, -9.52, 0.48]}