Step by step
The question
Question: find the maximum and minimum points of f(x) = x² − 6x + 2 with the second derivative test.
There are four choices.
First we solve it, the normal way.
Then we pick.
1 Find f′(x)
Use the power rule on each term.
Now we have f′(x) = 2x − 6.
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.
That is 2x − 6 = 0.
To remove the − 6, we add 6 on both sides.
To remove times 2, we divide both sides by 2.
6 divided by 2 is 3.
So x = 3.
3 Find f″(x)
f double prime is the derivative of f′.
It tells how the slope changes.
Use the power rule again, now on f′.
Now we have f″(x) = 2.
If f″ is positive, the slope is growing: the curve bends up, like a cup.
Its bottom is a minimum.
If f″ is negative, it bends down, like a cap.
Its top is a maximum.
4 Test each point
At x = 3, f″ is 2: positive.
A cup.
So a minimum.
Careful!
Positive f″ is a cup.
The bottom of a cup is a minimum, not a maximum.
Its y: put 3 into f.
We get −7.
So: a minimum at (3, −7).
5 Check
Let's check with the graph.
A cup at the minimum.
Right!
6 Pick the choice
Our answer is min (3, −7).
That is choice A.
7 Why not the others
B is (−3, 29).
Wrong: solved 2x − 6 = 0 with the wrong sign.
C is (3, 2).
Wrong: put x = 3 into the 2 only.
D is (6, 2).
Wrong: set f(x) = 2, not f′(x) = 0.
Let's look back.
First: find f′(x).
Then: solve f′(x) = 0.
Then: find f″(x).
Then: test each point.
Then: check.
Then: pick the choice.
Last: why not the others.