My Math Forum 3 doubts- plz help!!

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 February 21st, 2012, 08:29 PM #1 Newbie   Joined: Jan 2012 Posts: 23 Thanks: 0 3 doubts- plz help!! Kindly help me with the following 3 questions and please tell me how you arrived at the answer. Thanks in advance Let f and g be two differentiable functions on (0, 1) such that f(0) = 2, f(1) = 6, g(0) = 0 and g(1) = 2. Then there exists ? ? (0, 1) such that f'(?) equals (a) (1/2)g'(?), (b) 2g'(?), (c) 6g'(?), (d) (1/6)g'(?). If f(1) = 0, f'(x) > f(x) for all x > 1, then f(x) is (a) positive valued for all x > 1, (b) negative valued for all x > 1, (c) positive valued on (1, 2) but negative valued on [2,?). (d) None of these. For all x, y ? (0, ?), a function f: (0, ?) --> R satisfies the inequality |f(x) - f(y)| ? cube(|x - y|). Then f is (a) an increasing function (b) a decreasing function (c) a constant function (d) None of these.
 February 22nd, 2012, 03:38 AM #2 Newbie   Joined: Jan 2012 Posts: 23 Thanks: 0 Re: 3 doubts- plz help!! Someone please answer!!
February 22nd, 2012, 05:18 AM   #3
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Re: 3 doubts- plz help!!

Quote:
 Originally Posted by vasudha Let f and g be two differentiable functions on (0, 1) such that f(0) = 2, f(1) = 6, g(0) = 0 and g(1) = 2. Then there exists ? ? (0, 1) such that f'(?) equals (a) (1/2)g'(?), (b) 2g'(?), (c) 6g'(?), (d) (1/6)g'(?).
Cauchy's mean value theorem:

$\frac{f'(\theta)}{g'(\theta)}\,=\,\frac{f( 1)\,-\,f(0)}{g(1)\,-\,g(0)}\,=\,2 \\
\Rightarrow\,f'(\theta)\,=\,2g'(\theta),\, g'(\theta)\,\ne\,0$

Quote:
 If f(1) = 0, f'(x) > f(x) for all x > 1, then f(x) is (a) positive valued for all x > 1, (b) negative valued for all x > 1, (c) positive valued on (1, 2) but negative valued on [2,?). (d) None of these.
$f'(1)\,\gt\,f(1)\,=\,0\,\Rightarrow\,f(x)\text { is positive valued for all }x\,>\,1$

 February 22nd, 2012, 07:13 AM #4 Newbie   Joined: Jan 2012 Posts: 23 Thanks: 0 Re: 3 doubts- plz help!! Thank you greg But I haven't got the 2nd question. We are just given that f'(x)>f(x) for all x>1.. how did you say that f'(1)>f(1)?? Please reply. And I would be very grateful to anyone who can help me with the 3rd question as well..
 February 22nd, 2012, 07:30 AM #5 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,806 Thanks: 1045 Math Focus: Elementary mathematics and beyond Re: 3 doubts- plz help!! I made an error on that one, my apologies.

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