Where is f(x) = x^3 - 3x^2 concave up Step by Step

Step by step

The question

Question: find where f(x) = x³ − 3x² is concave up and where it is concave down.

f(x) = x³ − 3x²

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) = 3x^(3 − 1) − 3 × 2x^(2 − 1)

Now we have f′(x) = 3x² − 6x.

= 3x² − 6x

Then f″: the power rule again, on f′.

f″(x) = 3 × 2x^(2 − 1) − 6 × 1

Now we have f″(x) = 6x − 6.

= 6x − 6

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

f″(x) = 0

6x − 6 = 0

Take out the 6.

6(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) = −6, f″(2) = 6

f″(0) is −6: negative.

So f bends down there: a cap.

cap

f″(2) is 6: 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 −2.

f(1) = 1³ − 3(1)² = 1 − 3 = −2

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

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