Post #6

Alrighty, the beginning of this week was great because I could actually do the homework, but then we got to optimization.

I really understand Rolle's Theorem, but first you have to know the Extreme Value Theorem which just states that a continuous function on a closed interval must have both a max or min on the interval. (They can occur at the endpoints.) Anyway, Rolle's Theorem gives the conditions that guarantee the existence of an extema in the interior of a closed interval.

You basically see if it continuous and differentiable between the interval. If not, you cannot use this theorem. (If no interval just use x-intercepts.) Then plug in both interval points. These must equal each or you cannot use this theorem. Then take the derivative, set it equal to 0 and solve for x, and you get your critical values. (make sure the points fall between the interval)

Example: f(x)=x^4-2x^2 on [-2,2] Find all values c for which f^1(c)=0
Is it continuous and differentiable? YES
f(-2)=(-2)^4-2(-2)^2=8
f(2)=(2)^4-2(2)^2=8
f(-2)=f(2)=8
4x^3-4x=0 4x(x^2-1)=0 x=0, 1, -1 c=0, 1, -1

Also, I understand Mean Value Theorem which just says if f 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^1(c)=f(b)-f(a)/b-a.

All you do is make sure it is continuous and differentiable between the interval. (if not you cannot use this theorem) Then find the slope using f^1(c)=f(b)-f(a)/b-a. Then take the derivative and set it equal to the slope and solve for x. This gives the critical points. (make sure the points fall between the interval)

Example: f(x)=5-4/x Find all values c on (1,4) such that f^1(c)=f(b)-f(a)/b-a.
Is it continuous? there is an infinite @ x=0 (which is not in the interval so you can continue with this theorem) Is it differentiable? not @ x=0 (which is also not on the interval)
f(4)=5-4/4=4 f(1)=5-4/1= 1 4-1/4-1 = 3/3=1
f^1(x)=4/x^2
4/x^2=1 x= + or - 2 (-2 is not on the interval) c=2

The only real problem is I'm having trouble with optimization. Is there any steps to follow? I just get confused on what equation to use and what to use from the problem. Each problem is sort of different which I guess is what is making me have trouble. And does anyone know when our next quiz is?