Using Rolles Theorem on F(x)=x^2+3x on [0,2]

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In summary, the conversation discusses the use of Rolle's theorem and the Mean Value Theorem to prove the existence of a local extremum for a continuous and differentiable function. The emphasis on these theorems may be misplaced, as they are considered trivial consequences of a deeper result about continuity. It is important to note that while the statement of Rolle's theorem may not imply the existence of a flex between two critical points, the stronger result does.
  • #1
physics_ash82
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Ok I think I can use Rolles theorem on F(x)=x^2+3x on the inteval [ 0,2]
because the derivative can be defined
so then I think I use the formula [f(b)-f(a)] / [b-a] to find f'(c), then set f'(c) = F'(x) and solve is this process right?:shy:
 
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  • #2
What are you even trying to prove?
 
  • #3
It seems you're using the mean value theorem, which is the one you listed. Rolle's theorem says that if f(a)=f(b) on some closed interval, then there must be some point c such that f'(c)=0.
 
  • #4
rolles theorem is a trivial consequence of the obvious (but deep) fact that a continuous function f which takes the same value twice must have a local extremum in between.

then rolle says if f is also differentiable, the derivative is zero there.

the MVT is then a further trivial consequence of rolle.

i.e. both rolle and MVT are trivial, but useful consequences of one deep result about continuity.

my point is that the emphasis on these two as big time theorems is quite misplaced.

even their statements take away something from the result, since just knowing the derivative is zero somewhere in between two points is decidedly weaker than knowing there is a local extremum.

for instance the statement of rolles theorem does not imply that between two critical points of a continuously differentiable function there must be a flex, but the stronger result does.
 
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  • #5
thanks for the replys

I had seen my error after my post I was using the mean value theorem instead of rolles theorem opps:blushing: thanks for the replys though
 

1. What is Rolle's Theorem?

Rolle's Theorem is a mathematical theorem that states that if a function is continuous on a closed interval and differentiable on the open interval, and if the values of the function at the endpoints of the interval are equal, then there exists at least one point within the interval where the derivative of the function is equal to 0.

2. How do you use Rolle's Theorem on a given function?

To use Rolle's Theorem on a function, you need to make sure that the function satisfies the conditions of the theorem. This includes being continuous on a closed interval and differentiable on the open interval. Once these conditions are met, you can find the values of the function at the endpoints of the interval and set them equal to each other. Then, you can take the derivative of the function and set it equal to 0 to solve for the point where the derivative is equal to 0.

3. What is the given function in this case?

The given function is F(x) = x^2 + 3x on the interval [0,2].

4. How do you find the values of the function at the endpoints of the interval?

To find the values of the function at the endpoints of the interval, you simply plug in the values of the endpoints into the function. In this case, the endpoints are 0 and 2, so we would plug in 0 and 2 for x in the function F(x) = x^2 + 3x to get F(0) = 0 and F(2) = 10.

5. What does it mean when the derivative of a function is equal to 0 at a certain point?

When the derivative of a function is equal to 0 at a certain point, it means that the slope of the tangent line at that point is 0. This can also indicate a local maximum or minimum point on the graph of the function.

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