Step by step
The question
Question: find the concavity and the inflection points of f(x) = x⁴ − 4x³.
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.
1 Find f″(x)
First f′: the power rule on each term.
Now we have f′(x) = 4x³ − 12x².
Then f″: the power rule again, on f′.
Now we have f″(x) = 12x² − 24x.
2 Solve f″(x) = 0
The bending can only change where f″ is 0.
So we solve f″(x) = 0.
Every term has 12x: take it out.
A product is 0 only when one of its parts is 0.
So x = 0 or x = 2.
3 Sign table for f″
A sign table for f″.
The numbers 0 and 2 cut the line into 3 parts.
Test one number in each part: −1, 1, 3.
f″(−1) is 36: positive.
So f bends up there: a cup.
f″(1) is −12: negative.
So f bends down there: a cap.
f″(3) is 36: positive.
So f bends up there: a cup.
Careful!
Concave up is about the bending, not about going up.
A cup can go down on one side.
4 Inflection points
At x = 0, the bending changes: an inflection point.
At x = 2, the bending changes: an inflection point.
Its y: put 0 into f.
We get 0.
Its y: put 2 into f.
We get −16.
f is concave up where f″ is positive: x < 0 and x > 2.
f is concave down where f″ is negative: 0 < x < 2.
And the inflection points are (0, 0) and (2, −16).
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.