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March 15th, 2011, 09:11 AM   #1
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Combination of knowledge!

[color=#000000]1. Let A(1,0), B(0,1) and ?(1,1) three points on the 2D plane. For a random point M, which lies on the line segment AB without it being point A or B, prove that (? is the rate of direction and O is the beginning of the axis)

2. If a function f is continuous on the interval [0,1] and differentiable on (0,1) with f(0)=0 and f(1)=1 prove that there exist and with such that .[/color]
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March 17th, 2011, 09:14 PM   #2
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Re: Combination of knowledge!

Intermediate value theorem, mean value theorem?
No-one else going to touch this? ("? is the rate of direction" means "? is the slope.")
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March 24th, 2011, 04:53 PM   #3
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Re: Combination of knowledge!

[color=#000000]Aswoods is right, noone is going to touch this?[/color]
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March 25th, 2011, 02:54 AM   #4
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Re: Combination of knowledge!

1.
We have |OB|=1, |A?|=1, |OA|=1 and |B?|=1 (and data provided by ZardoZ)
The line segment is described by y=1-x
Let M have coördinates (x,y) such that it has coördinates (x,1-x).
Let's find slope
It is for
Now, we need slope as well.
It is for

Now, for

2. Not sure for this one.
Maybe, split in 3 cases.
C1: f'(0)<1. Then there exists f'(x)>1 (for otherwise, )? such that
C2: f'(0)=1. Then there exists f'(x)=1 such that
C3: f'(0)>1. Then there exists f'(x)=1 (for otherwise, )?such that
Maybe, some-one is inspired now.
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March 25th, 2011, 06:01 AM   #5
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Re: Combination of knowledge!

Let g(x) = 1-x. Then f(0)=0, g(0)=1, but f(1)=1, g(1)=0. Use the intermediate value theorem to prove that the curves intersect for some x=a, then apply the mean value theorem to f over the intervals [0,a] and [a,1].
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March 26th, 2011, 10:22 AM   #6
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Re: Combination of knowledge!

[color=#000000]Well done![/color]
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