Make-up #5

Rolle's Theorem

To understand Rolle's Theorem, you must first understand the Extreme Value Theorem.  The Extreme Value Theorem states that a continuous function on the closed interval [a,b] must have both a maximum and a minimum.  They can be on the endpoint a and b though.

So Rolle's Theorem gives the conditions that guarantee the existence of an extrema in the interior of a closed interval.

Rolle's Theorem-If a function is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) and f(a)=f(b), then there is at least one point, designated "c", where the derivative of f(c)=0.

Example:  Find all points "c" where the derivative of f(c)=0  f(x)=x^4-2x^2   [-2,2]

1.  I first found that f(a) equals f(b).

(-2)^4-(-2)^2=8
(2)^4-2(2)^2=8

2.  Once finding that f(a)=f(b), I set the derivative equal to 0 and solved for x.

4x^3-4x=0
4x(x^2-1)=0
4x=0  x^2-1=0
x=-1,0,1

There are three point of extrema within the [-2,2] of the function x^4-2x^2.  One at x=-1  another at x=0 and the last at x=1.


Mean Value Theorem

The Mean Value Theorem states that If the function f(x) is continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a number "c" in (a,b) such that f'(c)=(f(b)-f(a))/(b-a).

This is essentially the same thing as average speed, which finds the slope of the equation.


Intermediate Value Theorem

The Intermediate Value Theorem states that if a function f(x) is continuous on [a,b] and K is any number between f(a) and f(b), then there is at least one number "c" where f(c)=K.