Use the second derivative test on f(x) = x^2 - 4x + 7

Step by step

The question

Question: find the maximum and minimum points of f(x) = x² − 4x + 7 with the second derivative test.

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.

Take out the 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: 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 2 into f.

We get 3.

So: a minimum at (2, 3).

5 Check

Let's check with the graph.

A cup at the minimum.

Right!

Let's look back.

First: find f′(x).

Then: solve f′(x) = 0.

Then: find f″(x).

Then: test each point.

Last: check.