Midterm 3

Question 1

Rolle’s Theorem

Answer 1

Let f be a function that satisfies the following three hypothesis:

1) f is continuous on the closed interval [a,b]
2) f is differentiable on the open interval (a,b)
3) f(a)=f(b)

Then there is a number c in (a,b) such that f’(c)=0.

Question 2

Mean Value Theorem

Answer 2

Let f be a function that satisfies the following hypothesis:

1) f is continuous on the closed interval [a,b]
2) f is differentiable on the open interval (a,b)

Then there is a number c in (a,b) such that: f’(c)=[f(b)-f(a)]/(b-a)

Question 3

Extreme Value Theorem

Answer 3

If f is a continuous function on a closed interval [a,b] then f has both a maximum and a minimum on the interval.