The function f(x) = -x^2 + 3x satisfies the Mean Value Theorem on… Step by Step

Step by step

The question

Question: find the number c from the mean value theorem for f(x) = −x² + 3x on [−1, 2].

f(x) = −x² + 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

There are four choices.

First we solve it, the normal way.

Then we pick.

1 Check the conditions

f is a polynomial.

Its graph has no breaks and no sharp corners.

So it is continuous on [−1, 2], and smooth between.

The theorem can be used.

a polynomial: no breaks, no corners ✓

The ends: f of −1 is −4.

f(−1) = −(−1)² + 3(−1) = −1 − 3 = −4

And f of 2 is 2.

f(2) = −2² + 3(2) = −4 + 6 = 2

2 The average slope

The average slope from −1 to 2: the change in f over the change in x.

It is 2.

f(2) − f(−1)2 − (−1) = 2 − (−4)2 − (−1) = 63 = 2

Careful!

Divide by the width, 2 minus −1.

The average slope is 2, not 6.

Careful: not f(2) − f(−1) = 6 ✗

3 Find c

The slope: use the power rule.

f′(x) = −2x^(2 − 1) + 3 × 1

Now we have f′(x) = −2x + 3.

= −2x + 3

At c the slope is the average slope: f′(c) = 2.

−2c + 3 = 2

Take away 3 on both sides, so they stay equal.

−2c = −1

Divide by −2: c = 1/2.

c = −1−2 = 12

4 Keep the ones inside

c = 1/2 is between −1 and 2.

Keep it.

c = 12 ≈ 0.5 between −1 and 2 ✓

So c = 1/2.

c = 12 ≈ 0.5

5 Check

Let's check: put c into f′.

We get 2, the average slope.

Right!

f′(12) = 2

And in a picture.

The line through the two ends has the average slope.

At c, the tangent line is parallel to it: the same slope.

Right!

6 Pick the choice

Our answer is c = 1/2 ≈ 0.5.

That is choice A.

A) 12

7 Why not the others

B is 3/2.

Wrong: solved f′(c) = 0: the slope of the chord is 2.

B) 32 ✗ solved f′(c) = 0: the slope of the chord is 2

C is 0.

Wrong: used the slope 3.

C) 0 ✗ used the slope 3

D is −1/2.

Wrong: lost the minus of −2c.

D) −12 ✗ lost the minus of −2c

Let's look back.

First: check the conditions.

Then: the average slope.

Then: find c.

Then: keep the ones inside.

Then: check.

Then: pick the choice.

Last: why not the others.

Answer