Blog #9

This week we started fresh with the 2nd nine weeks. 25% done school! We started and continued learning material from Chapter 3. At the beginning of the week we learned Rolle's Theorem and the Mean Value Theorem. I am going to explain how to use these the Rolle's Theorem.

Rolle's Theorem

Let f be continuous on the closed interval [a,b] and differentiable on the open interval (a,b).

If f(a) = f(b), then there is at least one number c in (a,b) such that f ' (c) = 0

*Test for continuity and differentiability before using Rolle's Theorem.

Rolle's Theorem tells if there is a max or a min on an interval, and typically, Rolle's Theorem uses x-intercepts.

Example:

Let f(x) = x^4 - 2x^2. Find all values of c in the interval (-2, 2) such f ' (c) = 0.

Step 1: Is the function continuous? Yes, because it is a polynomial.

Step 2: Is the function differentiable? Yes, because there are no points of non-differentiability or items in the function that make it not differentiable.

Step 3: Plug in the x and y from the open interval into the original equation and see if they both have the same output.

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

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

Now take the derivative of the original function and set it equal to zero. Then find the x-intercepts.

4x^3 - 4x = 0

4x(x^2 - 1) = 0

x = 0, -1, 1

By Rolle's Theorem, the function is continuous, differentiable, and f(-2) = f(2), then there must be at least one max or min on [-2, 2] where (f ' (x) = 0).