Mean value theorem for f(x) = x^3 + 2x on [-2, 2] Step by Step

Step by step

The question

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

f(x) = x³ + 2x

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, 2], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of −2 is −12.

f(−2) = (−2)³ + 2(−2) = −8 − 4 = −12

And f of 2 is 12.

f(2) = 2³ + 2(2) = 8 + 4 = 12

2 The average slope

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

It is 6.

f(2) − f(−2)2 − −2 = 12 − (−12)2 − (−2) = 244 = 6

Careful!

Divide by the width, 2 minus −2.

The average slope is 6, not 24.

Careful: not f(2) − f(−2) = 24 ✗

3 Find c

The slope: use the power rule.

f′(x) = 3x^(3 − 1) + 2 × 1

Now we have f′(x) = 3x² + 2.

= 3x² + 2

At c the slope is the average slope: f′(c) = 6.

3c² + 2 = 6

Take away 2 on both sides.

3c² = 4

Divide by 3.

c² = 43

Take the square root.

c is plus or minus 2 over √3, about 1.15.

c = 2√3 or −2√3

4 Keep the ones inside

c, about 1.15, is between −2 and 2.

Keep it.

c = 2√3 ≈ 1.15 between −2 and 2 ✓

c, about −1.15, is between −2 and 2.

Keep it.

c = −2√3 ≈ −1.15 between −2 and 2 ✓

So c = 2 over √3, about 1.15, and c = −2 over √3, about −1.15.

= c = 2√3 ≈ 1.15 and c = −2√3 ≈ −1.15

5 Check

Let's check: put c into f′.

We get 6, the average slope.

Right!

f′(1.1547) ≈ 6

Let's look back.

First: check the conditions.

Then: the average slope.

Then: find c.

Then: keep the ones inside.

Last: check.

Answer