Inflection points of f(x) = x^4 - 6x^2 + 5

Step by step

The question

Question: find the concavity and the inflection points of f(x) = x⁴ − 6x² + 5.

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 − 12x.

Then f″: the power rule again, on f′.

Now we have f″(x) = 12x² − 12.

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

Take out the 12.

x² − 1 is a difference of two squares: x² minus 1 squared.

So it is (x − 1)(x + 1).

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

So x = −1 or x = 1.

3 Sign table for f″

A sign table for f″.

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

Test one number in each part: −2, 0, 2.

f″(−2) is 36: positive.

So f bends up there: a cup.

f″(0) is −12: negative.

So f bends down there: a cap.

f″(2) is 36: 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 = −1, the bending changes: an inflection point.

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

Its y: put −1 into f.

We get 0.

Its y: put 1 into f.

We get 0.

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

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

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

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.