Other Theorems

Darboux’s Theorem

Let be differentiable on , and is strictly between and :

Proof Hint

Use and follow the proof pattern of IVT proof.

Rolle’s Theorem

Let be continuous on and differentiable on . And . Then:

Proof Hint

By Extreme Value Theorem, maximum and minimum exists for .

Consider cases:

1. Both minimum and maximum exist at and .
2. One of minimum or maximum occurs in .

Mean Value Theorem

Let be continuous on and differentiable on . Then:

Proof Hint

• Define
• will be equal to
• Use Rolle’s Theorem for

Cauchy’s Mean Value Theorem

Let and be continuous on and differentiable on , and Then:

Proof Hint

• Define
• will be equal to
• Use Rolle’s Theorem for

Mean value theorem can be obtained from this when .

L’Hopital’s Rule

Note

Be careful with the pronunciation.

• It’s not “Hospital’s Rule”, there are no “s”
• It’s not “Hopital’s Rule” either, there is a “L’”

Learn the correct pronounciation from this video on YouTube.

L’Hopital’s Rule can be used when all of these conditions are met. (here is some positive number). Select the appropriate range (as in the limit definition), say .

1. Either of these conditions must be satisfied
2. are continuous on (closed interval)
3. are differentiable on (open interval)
4. on (open interval)

Then:

Here, can be either a real number or . And it is valid for all types of “x limits”.

Proof Hint

L’Hopital’s rule can be proven using Cauchy’s Mean Value Theorem.