Where is f(x) = x^4 - 6x^2 concave up Step by Step

Step by step

The question

Question: find where f(x) = x⁴ − 6x² is concave up and where it is concave down.

f(x) = x⁴ − 6x²

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) = 4x^(4 − 1) − 6 × 2x^(2 − 1)

Now we have f′(x) = 4x³ − 12x.

= 4x³ − 12x

Then f″: the power rule again, on f′.

f″(x) = 4 × 3x^(3 − 1) − 12 × 1

Now we have f″(x) = 12x² − 12.

= 12x² − 12

2 Solve f″(x) = 0

The bending can only change where f″ is 0.

So we solve f″(x) = 0.

f″(x) = 0

12x² − 12 = 0

Take out the 12.

12(x² − 1) = 0

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

So it is (x − 1)(x + 1).

12(x + 1)(x − 1) = 0

a² − b² = (a − b)(a + b), b = 1

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

a × b = 0 → a = 0 or b = 0

So x = −1 or x = 1.

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) = 36, f″(0) = −12, f″(2) = 36

f″(−2) is 36: positive.

So f bends up there: a cup.

cup

f″(0) is −12: negative.

So f bends down there: a cap.

cap

f″(2) is 36: 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.

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

infl.

Its y: put −1 into f.

We get −5.

f(−1) = (−1)⁴ − 6(−1)² = 1 − 6 = −5

Its y: put 1 into f.

We get −5.

f(1) = 1⁴ − 6(1)² = 1 − 6 = −5

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

up ∪ : x < −1 and x > 1

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

down ∩ : −1 < x < 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.

Answer