Step by step
The question
Question: find the concavity and the inflection points of f(x) = x³ + 3x² − 2.
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 + 6x.
Then f″: the power rule again, on f′.
Now we have f″(x) = 6x + 6.
2 Solve f″(x) = 0
The bending can only change where f″ is 0.
So we solve f″(x) = 0.
That is 6x + 6 = 0.
Take away 6 on both sides, so they stay equal.
Divide by 6: x = −1.
3 Sign table for f″
A sign table for f″.
The number −1 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, 0.
f″(−2) is −6: negative.
So f bends down there: a cap.
f″(0) is 6: positive.
So f bends up there: a cup.
4 Inflection points
At x = −1, the bending changes: an inflection point.
Its y: put −1 into f.
We get 0.
f is concave up where f″ is positive: x > −1.
f is concave down where f″ is negative: x < −1.
And the inflection point is (−1, 0).
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 > −1; concave down: x < −1; inflection: (−1, 0).
That is choice D.
7 Why not the others
A is (0, −2).
Wrong: f′ = 0 there: a turning point.
B is (−2, 2).
Wrong: f′ = 0 there: the other turning point.
C is (1, 2).
Wrong: solved 6x + 6 = 0 with the wrong sign.
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.