Worked examples
The question
Question: find the concavity and the inflection points of f(x) = x³ − 3x².
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
0
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
And the inflection point is (1, −2).
inflection : (1, −2)
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 inflection : (1, −2)
▶ Watch it solved
The question
Question: find the concavity and the inflection points of f(x) = x³ − 6x² + 9x + 1.
f(x) = x3 − 6x2 + 9x + 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) = 3x^(3 − 1) − 6 × 2x^(2 − 1) + 9 × 1 + 0
Now we have f′(x) = 3x² − 12x + 9.
= 3x2 − 12x + 9
Then f″: the power rule again, on f′.
f″(x) = 3 × 2x^(2 − 1) − 12 × 1 + 0
Now we have f″(x) = 6x − 12.
= 6x − 12
2 Solve f″(x) = 0
The bending can only change where f″ is 0.
So we solve f″(x) = 0.
f″(x) = 0
6x − 12 = 0
Take out the 6.
6(x − 2) = 0
So x = 2.
x = 2
3 Sign table for f″
A sign table for f″.
The number 2 cuts the line into 2 parts.
Test one number in each part: 1, 3.
f″(1) = −6, f″(3) = 6
f″(1) is −6: negative.
So f bends down there: a cap.
cap
f″(3) is 6: positive.
So f bends up there: a cup.
cup
0
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 = 2, the bending changes: an inflection point.
infl.
Its y: put 2 into f.
We get 3.
f(2) = 23 − 6(2)2 + 9(2) + 1 = 8 − 24 + 18 + 1 = 3
f is concave up where f″ is positive: x > 2.
up ∪ : x > 2
f is concave down where f″ is negative: x < 2.
down ∩ : x < 2
And the inflection point is (2, 3).
inflection : (2, 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 inflection : (2, 3)
▶ Watch it solved
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
0
0
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 down ∩ : −1 < x < 1
▶ Watch it solved
