Find c for the mean value theorem, f(x) = x^3

Step by step

The question

Question: find the number c from the mean value theorem for f(x) = x³ − 3x on [−2, 3].

f(x) = x³ − 3x

The theorem: for some c between a and b, the slope at c equals the average slope from a to b.

f′(c) = f(b) − f(a)b − a

for some c between a and b

1 Check the conditions

f is a polynomial.

Its graph has no breaks and no sharp corners.

So it is continuous on [−2, 3], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of −2 is −2.

f(−2) = (−2)³ − 3(−2) = −8 + 6 = −2

And f of 3 is 18.

f(3) = 3³ − 3(3) = 27 − 9 = 18

2 The average slope

The average slope from −2 to 3: the change in f over the change in x.

It is 4.

f(3) − f(−2)3 − −2 = 18 − (−2)3 − (−2) = 205 = 4

Careful!

Divide by the width, 3 minus −2.

The average slope is 4, not 20.

Careful: not f(3) − f(−2) = 20 ✗

3 Find c

The slope: use the power rule.

f′(x) = 3x^(3 − 1) − 3 × 1

Now we have f′(x) = 3x² − 3.

= 3x² − 3

At c the slope is the average slope: f′(c) = 4.

3c² − 3 = 4

Add 3 on both sides.

3c² = 7

Divide by 3.

c² = 73

Take the square root.

c is plus or minus √7 over √3, about 1.53.

c = √7√3 or −√7√3

4 Keep the ones inside

c, about 1.53, is between −2 and 3.

Keep it.

c = √7√3 ≈ 1.53 between −2 and 3 ✓

c, about −1.53, is between −2 and 3.

Keep it.

c = −√7√3 ≈ −1.53 between −2 and 3 ✓

So c = √7 over √3, about 1.53, and c = −√7 over √3, about −1.53.

= c = √7√3 ≈ 1.53 and c = −√7√3 ≈ −1.53

5 Check

Let's check: put c into f′.

We get 4, the average slope.

Right!

f′(1.5275) ≈ 4

Let's look back.

First: check the conditions.

Then: the average slope.

Then: find c.

Then: keep the ones inside.

Last: check.

Answer