EXTREMELY important for the AP Exam Chapter 4 Theorems EXTREMELY important for the AP Exam

Rolle’s Theorem In the first quadrant, mark “a” and “b” on the x axis. Plot points at f(a) and f(b) such that f(a) = f(b) Connect the two points however you would like but NOT in a horizontal line. What do you notice?

Rolle’s Theorem f(a)=f(b) a b What do you see between a and b on each of these graphs????

Rolle’s Theorem f’(c) = 0 Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). If f(a) = f(b) then there is at least one number c in (a, b) at which f’(c) = 0

Mean Value Theorem This is a more generalized version of Rolle’s Theorem. First stated by Joseph-Louis Lagrange

Mean Value Theorem Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). There is some point c in (a, b) at which

Mean Value theorem a b

Mean Value Theorem Suppose that y = f(x) is continuous at every point of the closed interval [a, b] and differentiable at every point of its interior (a, b). There is some point c in (a, b) at which

Mean Value theorem a c b

Extreme Value Theorem If f is continuous on a closed interval [a, b] Then f attains both an absolute maximum M And an absolute minimum m In [a,b]

Extreme Value Theorem (cont) That is, there are numbers x1 and x2 in [a, b] With f(x1) = m and f(x2) = M And m ≤ f(x) ≤ M for every other x in [a, b]

y = f(x) M m b a

y = f(x) M m a b

y = f(x) M m a b