Inflection points of f(x) = x^4 - 2x^3

Step by step

The question

Question: find the concavity and the inflection points of f(x) = x⁴ − 2x³.

Concave up means the curve bends up, like a cup.

That is where f″ is positive.

Concave down bends down, like a cap: f″ is negative.

1 Find f″(x)

First f′: the power rule on each term.

Now we have f′(x) = 4x³ − 6x².

Then f″: the power rule again, on f′.

Now we have f″(x) = 12x² − 12x.

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

Every term has 12x: take it out.

A product is 0 only when one of its parts is 0.

So x = 0 or x = 1.

3 Sign table for f″

A sign table for f″.

The numbers 0 and 1 cut the line into 3 parts.

Test one number in each part: −1, 0.5, 2.

f″(−1) is 24: positive.

So f bends up there: a cup.

f″(0.5) is −3: negative.

So f bends down there: a cap.

f″(2) is 24: positive.

So f bends up there: a cup.

Careful!

Concave up is about the bending, not about going up.

A cup can go down on one side.

4 Inflection points

At x = 0, the bending changes: an inflection point.

At x = 1, the bending changes: an inflection point.

Its y: put 0 into f.

We get 0.

Its y: put 1 into f.

We get −1.

f is concave up where f″ is positive: x < 0 and x > 1.

f is concave down where f″ is negative: 0 < x < 1.

And the inflection points are (0, 0) and (1, −1).

5 Check

Let's check with the graph.

A cup, then a cap, then a cup.

It changes at the inflection points.

Right!

Let's look back.

First: find f″(x).

Then: solve f″(x) = 0.

Then: sign table for f″.

Then: inflection points.

Last: check.