March 15th, 2011, 09:11 AM  #1 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  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] 
March 17th, 2011, 09:14 PM  #2 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Combination of knowledge!
Intermediate value theorem, mean value theorem? Noone else going to touch this? ("? is the rate of direction" means "? is the slope.") 
March 24th, 2011, 04:53 PM  #3 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Combination of knowledge! [color=#000000]Aswoods is right, noone is going to touch this?[/color] 
March 25th, 2011, 02:54 AM  #4 
Math Team Joined: Apr 2010 Posts: 2,780 Thanks: 361  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=1x Let M have coördinates (x,y) such that it has coördinates (x,1x). 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, someone is inspired now. 
March 25th, 2011, 06:01 AM  #5 
Senior Member Joined: Feb 2009 From: Adelaide, Australia Posts: 1,519 Thanks: 3  Re: Combination of knowledge!
Let g(x) = 1x. 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].

March 26th, 2011, 10:22 AM  #6 
Math Team Joined: Nov 2010 From: Greece, Thessaloniki Posts: 1,990 Thanks: 133 Math Focus: pre pre pre pre pre pre pre pre pre pre pre pre calculus  Re: Combination of knowledge! [color=#000000]Well done![/color] 

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