Step by step
The question
Question: find the maximum and minimum points of f(x) = −x² − 6x 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: 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 −3 into f.
We get 9.
So: a maximum at (−3, 9).
5 Check
Let's check with the graph.
A cap at the maximum.
Right!
6 Pick the choice
Our answer is max (−3, 9).
That is choice A.
7 Why not the others
B is (3, −27).
Wrong: solved −2x − 6 = 0 with the wrong sign.
C is (−3, −9).
Wrong: put in −3 and lost a sign: −(−3)² is −9, −6(−3) is +18.
D is (0, 0).
Wrong: where f is 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.