Step by step
The question
Question: find the number c from the mean value theorem for f(x) = x³ − x on [0, 2].
The theorem: for some c between a and b, the slope at c equals the average slope from a to 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 [0, 2], and smooth between.
The theorem can be used.
The ends: f of 0 is 0.
And f of 2 is 6.
2 The average slope
The average slope from 0 to 2: the change in f over the change in x.
It is 3.
Careful!
Divide by the width, 2 minus 0.
The average slope is 3, not 6.
3 Find c
The slope: use the power rule.
Now we have f′(x) = 3x² − 1.
At c the slope is the average slope: f′(c) = 3.
Add 1 on both sides.
Divide by 3.
Take the square root.
A plus number and a minus number both square to it.
So c is plus or minus 2 over √3, about 1.15.
4 Keep the ones inside
c, about 1.15, is between 0 and 2.
Keep it.
c, about −1.15, is not between 0 and 2.
It does not count.
So c = 2 over √3, about 1.15.
5 Check
Let's check: put c into f′.
We get 3, the average slope.
Right!
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 = 2 over √3 ≈ 1.15.
That is choice D.
7 Why not the others
A is 1.
Wrong: the middle of the interval: c must give f′(c) = 3.
B is 4/3.
Wrong: that is c²: take its square root.
C is 2.
Wrong: an end of the interval: c is inside it.
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.