Concavity and inflection points of f(x) = 2x^3

Step by step

The question

Question: find the concavity and the inflection points of f(x) = 2x³ − 6x² + 1.

f(x) = 2x³ − 6x² + 1

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.

cup concave up: f″ > 0

cap concave down: f″ < 0

1 Find f″(x)

First f′: the power rule on each term.

f′(x) = 2 × 3x^(3 − 1) − 6 × 2x^(2 − 1) + 0

Now we have f′(x) = 6x² − 12x.

= 6x² − 12x

Then f″: the power rule again, on f′.

f″(x) = 6 × 2x^(2 − 1) − 12 × 1

Now we have f″(x) = 12x − 12.

= 12x − 12

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

f″(x) = 0

12x − 12 = 0

Take out the 12.

12(x − 1) = 0

So x = 1.

x = 1

3 Sign table for f″

A sign table for f″.

The number 1 cuts the line into 2 parts.

Test one number in each part: 0, 2.

f″(0) = −12, f″(2) = 12

f″(0) is −12: negative.

So f bends down there: a cap.

cap

f″(2) is 12: positive.

So f bends up there: a cup.

cup

Careful!

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

A cup can go down on one side.

Careful: concave up is not "going up"

4 Inflection points

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

infl.

Its y: put 1 into f.

We get −3.

f(1) = 2(1)³ − 6(1)² + 1 = 2 − 6 + 1 = −3

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

up ∪ : x > 1

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

down ∩ : x < 1

And the inflection point is (1, −3).

inflection : (1, −3)

5 Check

Let's check with the graph.

A cap, then a cup.

It changes at the inflection point.

Right!

Let's look back.

First: find f″(x).

Then: solve f″(x) = 0.

Then: sign table for f″.

Then: inflection points.

Last: check.

Answer