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So how exactly do you do this?

Discussion in 'Mathematics' started by Maths_Mike, Feb 9, 2012.

  1. Maths_Mike

    Maths_Mike New commenter

    Upset one of my years 12's when I couldnt exaplain how to do this. Searched on google and it seems I am not the only one.
    f '(X) is x^2 -2 + 1/x^2 prove that f(x) is an increasing function (Edexcel P1 June 01 Q5)
    Ofcourse I get the fact that f '(X) >0 for an increasing function but given that
    f '(X) is x^2 -2 + 1/x^2 how do you get to (x-1/x)^2
    (Knowing the answer its obvious that (x-1/x)^2 is identical to x^2 -2 + 1/x^2 and that its alwasy positive I just cant do the completing the square bit)
     
  2. bombaysapphire

    bombaysapphire Star commenter

    Is there an x missing in your post?
    If f'(x) = (x^2-2x+1)/x^2 then it is equal to [(x-1)/x]^2
    If it doesn't then why isn't it written as f'(x) = (x^2-1)/x^2 which clearly isn't positive if x^2<1!!
     
  3. For a quadratic of the form x^2 +bx+c the c term is half the b term squared for a perfect square, so in your example you can think of it as x^2 -(2/x)x + c and since the c term is1/(x^2) which is half the b term squared, you have a perferct square of (x-1/x)^2.
     
  4. bombaysapphire

    bombaysapphire Star commenter

    Apologies - I assumed brackets where there clearly are none.
     
  5. DM

    DM New commenter

    I'm minded to say it is obvious as I just look at the first and see the second but if some steps would help perhaps try this?
    x^2 - 2 + 1/x^2 = (x^4 - 2x^2 + 1) / x^2 = (x^2 -1)^2 / x^2 = (x - 1/x)^2
     
  6. Shouldn't the last bit simplify to (x^2-1/x)^2 ? Perhaps that is the error that Mike has spotted?
     
  7. DM

    DM New commenter

    No?
     
  8. Maths_Mike

    Maths_Mike New commenter

    No error I am just not as good as DM and couldnt see it - thanks DM
     
  9. Next time I'll put my reading glasses on before daring to suggest that DM has made a mistake! I was reading it as (x-1)/x instead of x-1/x. Perhaps having a drink after browsing the forum instead of before would help as well:)
     

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