Step by step
The question
Question: find the maximum and minimum points of f(x) = −x² + 4x + 1 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 + 4.
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 + 4 = 0.
To remove the + 4, we take away 4 on both sides.
To remove times −2, we divide both sides by −2.
−4 divided by −2 is 2.
So x = 2.
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 = 2, f″ is −2: negative.
A cap.
So a maximum.
Careful!
Negative f″ is a cap.
The top of a cap is a maximum, not a minimum.
Its y: put 2 into f.
We get 5.
So: a maximum at (2, 5).
5 Check
Let's check with the graph.
A cap at the maximum.
Right!
6 Pick the choice
Our answer is max (2, 5).
That is choice B.
7 Why not the others
A is (−2, −11).
Wrong: solved −2x + 4 = 0 with the wrong sign.
C is (2, −3).
Wrong: lost the + 4x when putting in x = 2.
D is (4, 1).
Wrong: set f(x) = 1, 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.