Step by step
The question
Question: find the number c from the mean value theorem for f(x) = x² + 2x on [0, 4].
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, 4], and smooth between.
The theorem can be used.
The ends: f of 0 is 0.
And f of 4 is 24.
2 The average slope
The average slope from 0 to 4: the change in f over the change in x.
It is 6.
Careful!
Divide by the width, 4 minus 0.
The average slope is 6, not 24.
3 Find c
The slope: use the power rule.
Now we have f′(x) = 2x + 2.
At c the slope is the average slope: f′(c) = 6.
Take away 2 on both sides, so they stay equal.
Divide by 2: c = 2.
4 Keep the ones inside
c = 2 is between 0 and 4.
Keep it.
So c = 2.
5 Check
Let's check: put c into f′.
We get 6, the average slope.
Right!
6 Pick the choice
Our answer is c = 2.
That is choice B.
7 Why not the others
A is 6.
Wrong: that is the slope of the chord: solve f′(c) = 6.
C is 4.
Wrong: an end of the interval: c is inside it.
D is 3.
Wrong: solved 2c = 6: lost the + 2 of f′(c) = 2c + 2.
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.