Finding Point(s) of Inflection for a Cubic Function - Explanation and Solution

[katie2142]

Homework Statement

find the x-coordinate(s) of the point(s) of inflection of the function f(x) = x^3 - x^2 - x +1. Give valid reasons for your answer(s).

Homework Equations

The Attempt at a Solution

I don't quite know where to start. Do I somehow put them in my calculator?

 

[stunner5000pt]

You find the points of inflection by finding the second derivative and plugging that equal to zer oand solve for x.

 

[Feldoh]

More importantly, do you know what an inflection point is?

 

[katie2142]

Visually, I know what a point of inflection is, but I don't really know how to define it or how it relates to the second derivative. how does it?

but what you said rings a bell. so the second derivative is 6x -2. set equal to zero you get x = 1/3. so is that the x coordinate? if i wanted to find the y coordinate, i would plug 1/3 into my original f(x)?

 

[HallsofIvy]

A 'point of inflection' is a point at which the graph changes concavity: from "concave up to concave down" or vice-versa. Of course (if you are expected to do problems like this, it should be "of course"!) the graph is "concave up" where the second derivative is positive and "concave down" where the second derivative is negative. That means that the second derivative must change sign there and can only do that where the second derivative is 0.

Be careful: the problem specifically asks you to "give valid reasons for your answer"! While the second derivative must be 0 at a point of inflection, the converse is not true- for example, f(x)= x⁴ has second derivative f"(x)= 12x² which is 0 at x= 0. But it is easy to see that the graph is always "concave up" and so there are NO points of inflection.

Here it is true that the second derivative changes sign only at x= 1/3. Does the second derivative actually change sign there?
(The original problem only asked for the x-coordinate of a point of inflection. I see no reason for finding the y-coordinate there.)

(And don't ever say "Do I somehow put them in my calculator?". You are supposed to be smarter than a calculator!)