Use the second derivative test on f(x) = x^3 + 1

Step by step

The question

Question: find the maximum and minimum points of f(x) = x³ + 1 with the second derivative test.

1 Find f′(x)

Use the power rule on each term.

Now we have f′(x) = 3x².

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.

3x² is 0 only when x is 0.

So x = 0.

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) = 6x.

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 = 0, f″ is 0.

The test cannot tell.

So we look at the slope on each side, as in the max and min lesson.

Up before, up after.

It does not turn: not a max, not a min.

So there is no maximum or minimum point.

5 Check

Let's check with the graph.

It never turns.

Right!

Let's look back.

First: find f′(x).

Then: solve f′(x) = 0.

Then: find f″(x).

Then: test each point.

Last: check.