Mean value theorem for f(x) = 2x^2 - x on [1, 4] Step by Step

Step by step

The question

Question: find the number c from the mean value theorem for f(x) = 2x² − x on [1, 4].

f(x) = 2x² − x

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 [1, 4], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of 1 is 1.

f(1) = 2(1)² − 1 = 2 − 1 = 1

And f of 4 is 28.

f(4) = 2(4)² − 4 = 32 − 4 = 28

2 The average slope

The average slope from 1 to 4: the change in f over the change in x.

It is 9.

f(4) − f(1)4 − 1 = 28 − 14 − 1 = 273 = 9

Careful!

Divide by the width, 4 minus 1.

The average slope is 9, not 27.

Careful: not f(4) − f(1) = 27 ✗

3 Find c

The slope: use the power rule.

f′(x) = 2 × 2x^(2 − 1) − 1

Now we have f′(x) = 4x − 1.

= 4x − 1

At c the slope is the average slope: f′(c) = 9.

4c − 1 = 9

Add 1 on both sides, so they stay equal.

4c = 10

Divide by 4: c = 5/2.

c = 104 = 52

4 Keep the ones inside

c = 5/2 is between 1 and 4.

Keep it.

c = 52 ≈ 2.5 between 1 and 4 ✓

So c = 5/2.

= c = 52 ≈ 2.5

5 Check

Let's check: put c into f′.

We get 9, the average slope.

Right!

f′(52) = 9

Let's look back.

First: check the conditions.

Then: the average slope.

Then: find c.

Then: keep the ones inside.

Last: check.

Answer