Three Big Theorems (8.1.2)

Question 1

• The intermediate value theorem: If f is continuous on a closed interval [a, b] and Q is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = Q.

Answer 1

• The intermediate value theorem: If f is continuous on a closed interval [a, b] and Q is any number between f(a) and f(b), then there is at least one number c in [a, b] such that f(c) = Q.

Question 2

• Rolle’s theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is at least one number c in (a, b) such that f (c) = 0.

Answer 2

• Rolle’s theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there is at least one number c in (a, b) such that f (c) = 0.

Question 3

• The mean value theorem: If f is continuous on [a, b] and differentiable on (a, b), then there is at least one number c in (a, b) such that (f(b) - f(a))/(b-a) = f’(c).

Answer 3

• The mean value theorem: If f is continuous on [a, b] and differentiable on (a, b), then there is at least one number c in (a, b) such that (f(b) - f(a))/(b-a) = f’(c).