Step by step
The question
Question: find the number c from Rolle's theorem for f(x) = x³ − 4x on [−2, 2].
The theorem: if f is the same at both ends, then for some c between them, the slope is 0.
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.
The ends: f of −2 is 0.
And f of 2 is 0.
Both are 0.
The same height, so Rolle's theorem works here.
2 Find c
The slope: use the power rule.
Now we have f′(x) = 3x² − 4.
At c the slope is 0.
So f′(c) = 0.
Add 4 on both sides.
Divide by 3.
Take the square root.
c is plus or minus 2 over √3, about 1.15.
3 Keep the ones inside
c, about 1.15, is between −2 and 2.
Keep it.
c, about −1.15, is between −2 and 2.
Keep it.
So c = 2 over √3, about 1.15, and c = −2 over √3, about −1.15.
4 Check
Let's check: put c into f′.
We get 0: flat.
Right!
And in a picture.
The two ends are at the same height.
At c, the curve is flat: its tangent line is level.
Right!
Let's look back.
First: check the conditions.
Then: find c.
Then: keep the ones inside.
Last: check.