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Math Help - prove or desprove claims

  1. #1
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    prove or desprove claims

    determine if the following are correct or wrong
    if correct prove if not then show counter example:
    a)if lim_{x->\infty}f(x)=\infty and g(x)>0 for x\in R then
    lim_{x->\infty}f(x)g(x)=\infty
    couldnt find counter example ,i thought f(x)=x but there is not g(x)
    to desprove it

    b) 2xArctanx>ln(1+x^{2}) for x>0
    i was told told to use the theore, of lagrange
    in which the derivative in some point is the slope between two points
    f'(c)=\frac{f(b)-f(a)}{b-a} but i dont know how to use it

    c)f(x) if bound from the top on (0,1{]}
    if f(x) is continues from the left of x=1 then sup f((0,1))>=f(1)

    d)if f(x) is integrabile on {[}a,b{]} and if \intop_{a}^{b}f(x)dx>1
    then there is c\in(a,b)
    so \intop_{a}^{b}f(x)dx=1
    Last edited by transgalactic; August 28th 2011 at 01:30 PM.
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  2. #2
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    Re: prove or desprove claims

    Quote Originally Posted by transgalactic View Post
    [LEFT]
    determine if the following are correct or wrong
    if correct prove if not then show counter example:
    a)if lim_{x->\infty}f(x)=\infty and g(x)>0 for x\in R then
    lim_{x->\infty}f(x)g(x)=\infty
    couldnt find counter example ,i thought f(x)=x but there is not g(x)
    to desprove it
    (a) how about f(x) = x and g(x) = \frac{1}{x^2+1} for a counter-example?
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  3. #3
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    Re: prove or desprove claims


    determine if the following are correct or wrong
    if correct prove if not then show counter example:
    a)if and for then

    couldnt find counter example ,i thought f(x)=x but there is not g(x)
    to desprove it
    b) for x>0
    i was told told to use the theore, of lagrange
    in which the derivative in some point is the slope between two points
    but i dont know how to use it
    c)f(x) if bound from the top on (0,1{]}
    if f(x) is continues from the left of x=1 then sup f((0,1))>=f(1)
    d)if f(x) is integrabile on {[}a,b{]} and if
    then there is
    so

    (a)
    I like your idea. How about f(x)=x, g(x)=1/x^2 ?

    (d)
    is the following function continuous? g(c) = \int_a^c f(x) dx

    if so you can use the intermediate value theorum, as its trivial to show that there are values of g(c) on either side of 1.
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  4. #4
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    Re: prove or desprove claims

    yes thanks
    1 down 3 to go
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  5. #5
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    Re: prove or desprove claims

    regarding b) i was told to turn 2xArctanx>ln(1+x^{2}) into 2xArctanx-ln(1+x^{2})>0
    f(x)=2xArctanx-ln(1+x^{2})
    so i need to prove that f(x)>0 for x>0
    f(0)=0
    now i need to show that f'>0 so f will be monotonicly increasing
    f'(x)=2Arctanx+2\frac{1}{1+x^2}- \frac{2x}{1+x^{2}}
    how to conclude that f' is positive for x>0
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