# What is the difference between rolle's theorem and mean value theorem?

Rolle's Theorem is a special case of the Mean Value Theorem.

Difference 1 Rolle's theorem has 3 hypotheses (or a 3 part hypothesis), while the Mean Values Theorem has only 2.

Difference 2 The conclusions look different.

The difference really is that the proofs are simplest if we prove Rolle's Theorem first, then use it to prove the Mean Value Theorem. After we have done that, we don't really need Rolle's Teorem for anything else. I mean, every other use (other than proving Mean Value) for Rolle's Theorem can be handled by the Mean Value Theorem.

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Rolle's Theorem states that if a function ( f(x) ) is continuous on a closed interval ([a, b]), differentiable on the open interval ((a, b)), and ( f(a) = f(b) ), then there exists at least one point ( c ) in the interval ((a, b)) such that ( f'(c) = 0 ).

Mean Value Theorem states that if a function ( f(x) ) is continuous on a closed interval ([a, b]) and differentiable on the open interval ((a, b)), then there exists at least one point ( c ) in the interval ((a, b)) such that ( f'(c) ) is equal to the average rate of change of ( f(x) ) over ([a, b]), which is ( \frac{f(b) - f(a)}{b - a} ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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