Step by step
The question
Question: find where f(x) = x³ − 3x² is increasing and where it is decreasing.
The derivative is the slope.
Where it is positive, the graph goes up.
Where it is negative, the graph goes down.
1 Find f′(x)
Use the power rule on each term.
Now we have f′(x) = 3x² − 6x.
2 Solve f′(x) = 0
At a turning point the graph is flat for a moment: the slope is 0.
So we solve f′(x) = 0.
Take out the 3.
We need two numbers that multiply to 0 and add to −2: 0 and −2.
A product is 0 only when one of its parts is 0.
So x = 0 or x = 2.
Careful!
f′ = 0 alone does not tell max or min.
The signs on each side tell us.
3 Sign table
A sign table.
The numbers 0 and 2 cut the line into 3 parts.
Test one number in each part: −1, 1, 3.
f′(−1) is 9: positive.
So f goes up there.
f′(1) is −3: negative.
So f goes down there.
f′(3) is 9: positive.
So f goes up there.
4 Read the table
f is increasing, going up, where f′ is positive: x < 0 and x > 2.
f is decreasing, going down, where f′ is negative: 0 < x < 2.
5 Check
Let's check with the graph.
Up, then down, then up: just like the table.
Right!
Let's look back.
First: find f′(x).
Then: solve f′(x) = 0.
Then: sign table.
Then: read the table.
Last: check.