Step by step
The question
Question: find the concavity and the inflection points of f(x) = x³ − 9x² + 24x.
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) = 3x² − 18x + 24.
Then f″: the power rule again, on f′.
Now we have f″(x) = 6x − 18.
2 Solve f″(x) = 0
The bending can only change where f″ is 0.
So we solve f″(x) = 0.
Take out the 6.
So x = 3.
3 Sign table for f″
A sign table for f″.
The number 3 cuts the line into 2 parts.
Test one number in each part: 2, 4.
f″(2) is −6: negative.
So f bends down there: a cap.
f″(4) is 6: 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 = 3, the bending changes: an inflection point.
Its y: put 3 into f.
We get 18.
f is concave up where f″ is positive: x > 3.
f is concave down where f″ is negative: x < 3.
And the inflection point is (3, 18).
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.