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.
Concave down bends down, like a cap.
f″ is the slope of the slope.
When f″ is positive, the slope keeps growing, so the curve turns up: a cup.
When f″ is negative: a cap.
Careful!
Concave up is about the bending, not about going up.
A cup can go down on one side.
There are four choices.
First we solve it, the normal way.
Then we pick.
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.
That is 6x − 18 = 0.
Add 18 on both sides, so they stay equal.
Divide by 6: x = 3.
3 Sign table for f″
A sign table for f″.
The number 3 cuts the line into 2 parts.
In each part, f″ keeps one sign.
So one test number per part is enough.
Take easy ones: 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.
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!
6 Pick the choice
Our answer is concave up: x > 3; concave down: x < 3; inflection: (3, 18).
That is choice A.
7 Why not the others
B is (2, 20).
Wrong: f′ = 0 there: that is a turning point.
C is (4, 16).
Wrong: f′ = 0 there: the other turning point.
D is (0, 0).
Wrong: where f is 0.
Let's look back.
First: find f″(x).
Then: solve f″(x) = 0.
Then: sign table for f″.
Then: inflection points.
Then: check.
Then: pick the choice.
Last: why not the others.