Other Theorems

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 .

Generalized MVT for Riemann Integrals

Let be continuous on ( are integrable), and does not change sign on . Then such that:

Proof Hint

• Use Extreme value theorem for
• Multiply by . Then integrate. Then divide by .
• Use intermediate value theorem to find

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’”

L’Hopital’s Rule can be used when all of these conditions are met. (here is some positive number). Select the appropriate ranges.

1. Either of these conditions must be satisfied
2. are continuous on
3. are differentiable on
4. on

Then:

Note

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

It is valid for all types of “x limits”.