Determine why the statement below is not a good enough description of the Mean V

Question

Determine why the statement below is not a good enough description of the Mean Value Theorem.

If there is a value c on the interval (a,b) where f'(c)=[f(b)-f(a)]/(b-a), then the function f(x) must be continuous on [a,b] and differentiable on (a,b).

I am aware that the Mean Value Theorem states that if f(x) is defined and continuous on the interval [a,b] and differentiable on (a,b), then there is at least one number c in the interval (a,b) such that f'(c)=[f(b)-f(a)]/(b-a). Why is the above statement not a good enough description?

Explanation / Answer

It's not a good enough description because of the first phrase

stating if there is a value of c in the interval (a, b), etc.

The mean value confirms that there exists c when f(x) is continuous and f'(x) exists.

The statement above does not give a property of a diffferentiable and continuous function

but rather describes a condition in a which a function is differentiable and continuous.

The key is that the mean value theorem guarantees the existence of c

while the latter doesn't.

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