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Math Help - Stationary points, local maxima, minima of a function

  1. #1
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    Stationary points, local maxima, minima of a function

    Hi,

    Does anyone know how you work out the intervals on which a function is increasing or decreasing and any stationary points, local maxima and local minima?

    The function I have is f(x)=\frac{13-6x}{4-x^2}

    I have differentiated it and have f '(x)=-\frac{6x^2-26x+24}{(4-x^2)^2}

    Which can be factorised to \frac{2(-3x-4)(x-3)}{(2+x)(2-x)}

    From other eg's I think a sign table is the best way of doing this, but I'm not sure what the headings / columns should be.

    From trial and error I know there is a stationary point at 3, i.e. f '(x) is 0 when x=3, but I'm not sure how to work out the other things.

    I would be really grateful for any advice
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  2. #2
    Senior Member yeKciM's Avatar
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    your function is defined on  (-\infty, -2) \cup (2 , + \infty)
    that u'll know from  4-x^2	\neq0

    your zero point is  13-6x=0 \Rightarrow x=\frac {13}{6}

    there are like 7 thing's for u to do ti'll you are able to draw that function ....
    u need first and second derivation .... and so on
    are u familiar how (and why ) each of it is done ?

    Edit :: lost while translate sorry (hello Ackbeet and thanks )



    Last edited by yeKciM; August 2nd 2010 at 12:49 PM.
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  3. #3
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    I disagree. The function is defined on \mathbb{R}\setminus\{-2,2\}. Stationary points are where the derivative is equal to zero. In setting a fraction equal to zero, you need only make sure that at those points, the numerator is zero, and the denominator is not zero. Use the second derivative test to find maxima and minima.
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  4. #4
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    Quote Originally Posted by yeKciM View Post
    your function is defined on  (-\infty, -2) \cup (2 , + \infty)
    that u'll know from  4-x^2	\neq0

    your stationary point is  13-6x=0 \Rightarrow x=\frac {13}{6}

    there are like 7 thing's for u to do ti'll you are able to draw that function ....
    u need first and second derivation .... and so on
    are u familiar how (and why ) each of it is done ?
    Hey. Thanks for the reply

    I have a strategy in my book I have been working from:
    1. the domain of f (all real numbers except +/- 2)
    2. whether f is odd or even (f is neither)
    3. the x and y intercepts of f (x=13/6, y=13/4)
    4. the intervals on which x is positive or negative (positive on (-2,0), 0, (0,2) and (2, \infty) and negative on (-\infty, -2)
    5. the intervals on which f is increasing / decreasing and any stationary points, local maxima, local minima - this is where I got stuck
    6. the asymptotic behaviour of f

    Is this what you meant? I haven't used the second derivative though. Do I need to?

    Thanks again
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  5. #5
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    Quote Originally Posted by Ackbeet View Post
    I disagree. The function is defined on \mathbb{R}\setminus\{-2,2\}. Stationary points are where the derivative is equal to zero. In setting a fraction equal to zero, you need only make sure that at those points, the numerator is zero, and the denominator is not zero. Use the second derivative test to find maxima and minima.
    On differentiating

    f '(x)=-\frac{6x^2-26x+24}{(4-x^2)^2} again, I get

    f ''(x)=-\frac{6x^3-10x^2+24x-16}{(4-x^2)^4}.

    Does this look like I've differentiated in correctly?
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  6. #6
    Senior Member yeKciM's Avatar
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    u need first derivation to find stationary points (sorry for that early) , then u find second derivation and when u equal it with zero u'll get saddle point's (sorry i didn't find better word to translate that ) if there are any
    but when u put your stationari point in second in second derivation u'll see if it is min or max when u put stationary point in second derivation and it's  f_{(x)} '' <0 \Rightarrow max ,or if it is   f_{(x)} '' >0 \Rightarrow min ...
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  7. #7
    Senior Member yeKciM's Avatar
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    Quote Originally Posted by cozza View Post
    On differentiating

    f '(x)=-\frac{6x^2-26x+24}{(4-x^2)^2} again, I get

    f ''(x)=-\frac{6x^3-10x^2+24x-16}{(4-x^2)^4}.

    Does this look like I've differentiated in correctly?
    \displaystyle f'(x)= \frac {-6x^2-26x-24}{(4-x^2)^2} \Rightarrow 3x^2-13x-12= -(3x+4)(x+3)=0 and you have your stationary points
    Last edited by yeKciM; August 2nd 2010 at 01:50 PM.
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  8. #8
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    Quote Originally Posted by yeKciM View Post
     f'(x)= \frac {-6x^2-26x-24}{(4-x^2)^2} \Rightarrow 3x^2-13x-12= -(3x+4)(x+3)=0 and you have your stationary points
    so f'(x) is 0 when x=3 or x=-4/3

    So if I now put those values of x into the second derivative if they are <0 then it is a local maxima and if >0 then it is a local minima?

    I got that x=3 is a local maxima and x=-4/3 a local minima

    For the asymptotic behaviour, since the denominator is 0 when x is +/- 2, I think the lines x=2 and x=-2 are vertical asymptotes. For horizontal asymptotes my book says "To detect horizontal asymptotes of a rational function, we divide bothe the numerator and denominator by the dominant term of the denominator and consider the behaviour as x\rightarrow\infty and as x\rightarrow-\infty.

    So, f(x)=\frac{13-6x}{4-x^2}=\frac{\frac{13}{x^2}-\frac{6x}{x^2}}{\frac{4}{x^2}-1}
    \rightarrow\frac{0-0}{0-1}=0 as x\rightarrow+/-\infty, which means f has the horizontal asymptote y=0, but that doesn't work, according to what I think the graph should look like?
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  9. #9
    Senior Member yeKciM's Avatar
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    hehehehe... i got lost there working on second so i can't confirm to u is it or is it not correct that one u have done sorry
    but there u are how your graphic should look like
    Attached Thumbnails Attached Thumbnails Stationary points, local maxima, minima of a function-function.jpg  
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  10. #10
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    That's what I get too. Thank you for all your help
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