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    The function f, defined for XER and x>0, is such that

    f ' (x) =x^"-2 + 1/x^2 yes that's f dash, i.e. gradient function or dy/dx



    (a) Find the value of f ''(4)

    (b) Given that f(3)=0, find f(x)

    (c) Prove that f is an increasing function

    any help is massively appreciated!
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    (a) Differentiate f'(x) and substitute x=4.
    (b) Integrate f'(x) remembering the constant of integration, then use f(3)=0 to find the required value of the constant.
    (c) You need to show f'(x) >= 0 for all x.
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    f ' (x) =x^"-2 + 1/x^2

    a) f'(x) = 1/x^(2) + 1/x^2
    = 1/4^(2) +1/(4)^2
    = 1/16+1/16
    = 2/16

    B) f(x) = integral of x^(-2) + x^(-2)
    = integral of 2x^(-2)
    = -2x^(-1) + C
    f(3) = 0
    -2(3)^(-1) + C = 0
    C = 2(3)^(-1)
    C = 1/3
    F(x) = -2x^(-1) + (1/3)
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    Sorry
    No,

    f ' (x) = x^2 -2 + 1/x^2

    in words: x sqaured, then minus two, then add 1 over x squared.
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    (Original post by darth_vader05)
    you can write that as f'(x) = x^2 + x^-2 -2

    differentiate that again and substitute 4 into it to get part a.
    Differenciate again??...he allready has F'(x) for part A. He doesn't need to do anything to it until part B where he integrates.
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    Part (c) is a little tricky. See if you can use the fact that

    (x^2-1)^2 >= 0 for all values of x
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    OHRIGHT sorry my mistake, didn't realise he was after F''(x)
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    Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

    Help!!
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    (Original post by Mush)
    Differenciate again??...he allready has F'(x) for part A. He doesn't need to do anything to it until part B where he integrates.
    Part (a) requires the second derivitive.
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    (Original post by Lawbutwhere?)
    Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

    Help!!
    See my post above.
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    dont really see how tht helps, ive got to show that

    x^2 -2 + 1/x^2 is always greater than zero.

    Sorry, probs just me...lol
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    Multiply it out, see what you get.
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    (Original post by Lawbutwhere?)
    Done parts (a) and (b) now, just need help with proving that f is an increasing function, i.e. the gradient is always >0

    Help!!
    increasing functionf(x) f'(x)>0 .
    x²-2+1/x²>0
    (x-1/x)²>0
    True for all x except f'(1)=0 .:. increasing function
 
 
 
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