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

Step by step

The question

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

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: negative.

So f bends down there: a cap.

cap

f″(0) is 12: positive.

So f bends up there: a cup.

cup

f″(2) is −36: negative.

So f bends down there: a cap.

cap

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: −1 < x < 1.

up ∪ : −1 < x < 1

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

down ∩ : x < −1 and x > 1

And the inflection points are (−1, 5) and (1, 5).

inflection : (−1, 5), (1, 5)

5 Check

Let's check with the graph.

A cap, then a cup, then a cap.

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