Step by step
The question
Question: find where f(x) = −x² + 2x + 3 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) = −2x + 2.
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 −2.
So x = 1.
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 1 cut the line into 2 parts.
Test one number in each part: 0, 2.
f′(0) is 2: positive.
So f goes up there.
f′(2) is −2: negative.
So f goes down there.
4 Read the table
f is increasing, going up, where f′ is positive: x < 1.
f is decreasing, going down, where f′ is negative: x > 1.
5 Check
Let's check with the graph.
Up, then down: 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.