The Intermediate Value Theorem guarantees a solution to [math]f(x)=2[/math] in which of the following intervals?

The Mean Value Theorem guarantees a solution to [math]f'(x)=1[/math] in which of the following intervals?

Give an interval over which the Extreme Value Theorem applies to [math]f[/math], but the Mean Value Theorem does not. Explain.

[math]f[/math] is defined and yet has no absolute maximum over the interval [math][4,6][/math]. Explain why this fails to contradict the Extreme Value Theorem.

Some calculus classes and textbooks name another theorem called Rolle's Theorem, usually stated as follows:[br][br]If [math]f[/math] is continuous over [math][a,b][/math] and differentiable over [math](a,b)[/math] and [math]f(a)=f(b)[/math], then there exists some [math]c[/math] in [math](a,b)[/math] such that [math]f'(c)=0[/math].[br][br]Rolle's Theorem is merely a special case of which of our theorems?