Extreme Value Theorem & MVT/Rolles Theorem

[carlodelmundo]

Homework Statement

If f is a continuous function on the closed interval [a,b], which of the following statements are NOT necessarily true?


I. f has a minimum on [a,b].

II. f has a maximum on [a,b].

III. f'(c) = 0 for some number c, a < c < b

Homework Equations

Extreme Value Theorem (EVT) - the EVT states that if f(x) is continuous on [a,b] there is one absolute maximum and one absolute minimum in [a,b].

The Attempt at a Solution

By the EVT... I believe statement "I" and statement "II" are always true.

Statement "III" is not necessarily true. The Mean Value Theorem/Rolles Theorem states that there is a c where f'(c) = 0 iff f(x) is continuous on [a,b] and iff f(x) is differentiable on (a,b). Since we weren't given the differentiability option... this is NOT necessarily true all the time.


Is this correct? Statement III is not necessarily true?

 

[quZz]

I, II are always true, III is not always true for example f(x) = x, [a,b] = [0,1], but you're wrong about Rolle's theorem.

 

[carlodelmundo]

Thank you qUzz!