Inflection points of f(x) = x^4

Step by step

The question

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

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.

Then f″: the power rule again, on f′.

Now we have f″(x) = 12x².

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

12x² is 0 only when x is 0.

So x = 0.

3 Sign table for f″

A sign table for f″.

The number 0 cuts the line into 2 parts.

Test one number in each part: −1, 1.

f″(−1) is 12: positive.

So f bends up there: a cup.

f″(1) is 12: 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, f″ is 0, but the sign does not change.

The bending stays the same: not an inflection point.

f is concave up where f″ is positive: all x.

There is no inflection point.

5 Check

Let's check with the graph.

A cup, then a cup.

Right!

Let's look back.

First: find f″(x).

Then: solve f″(x) = 0.

Then: sign table for f″.

Then: inflection points.

Last: check.