IARTV

[kamil9876]

yeah, u can use this to "prove" mean value theorem for differentiation. Do try

[/0]

OK!

Mean-value theorem: For at least one point on a curve, the gradient is equal to the average gradient over the smooth interval.

Say a function has average gradient in .

Then the angle of this gradient with respect to the positive x-axis is

If we rotate the graph clockwise, the average gradient is now zero, and the problem is equivalent to proving there is at least 1 stationary point - Rholle's theorem!

Rholle's theorem: For any two points with equal values on a smooth graph, there must be at least one stationary point between them.

(Honestly, I don't see why there is any need for a formal proof of Rholle's theorem... it's like proving the solution to "x = 1" is x = 1)

[TrueTears]

Rholle's theorem: For any two points with equal values on a smooth graph, there must be at least one stationary point between them.

I remember when reading about this, the line where satisfies the conditions, would that mean it has infinite stationary points?

EDIT: or on second thoughts, would it have none?

[/0]

I think it would have infinitely many stationary points lol... but then again, the definition of a stationary point is arbitrary

[TrueTears]

I think it would have infinitely many stationary points lol... but then again, the definition of a stationary point is arbitrary

haha, but for a stationary point doesn't the f''(x) also have to be defined? ie either larger than 0 or smaller than 0. In the case I provided you have 0 for f''(x) suggesting its a point of inflection? Eh maybe I'm just over complicating things but yeh