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Math Help - Here's another (harder) one

  1. #1
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    Here's another (harder) one

    Graph the following showing all working

    4X^3/((X-4)^2(X+2))

    The main problem i'm having with this one is finding the stationary point and the assymptotes. any help appreciated.
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  2. #2
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    Quote Originally Posted by mathsB View Post
    Graph the following showing all working

    4X^3/((X-4)^2(X+2))

    The main problem i'm having with this one is finding the stationary point and the assymptotes. any help appreciated.
    Hello,

    f(x)=\frac{4x^3}{(x-4)^2 \cdot (x+2)}. (the black curve)

    1. Domain: D = \mathbb{R} \setminus \{-2, 4\}. Thus the vertical asymptotes are: \boxed{x = 4~ or~ x = -2} . (the blue lines)

    2. Horizontal asymptote:

    \lim_{|x| \rightarrow \infty}f(x) = \lim_{|x| \rightarrow \infty}\left(\frac{4x^3}{(x-4)^2 \cdot (x+2)} \right) = \lim_{|x| \rightarrow \infty}\left(\frac{4x^3}{x^3-6x^2+32}\right) = 4. Thus the horizontal asymtote is: \boxed{y = 4} . (the red line)

    3. Zeros:
    f(x) = 0 if 4x = 0 that means there is one zero at x = 0.

    4. Stationary points. Calculate the 1st drivative of f. Use quotient rule:

    f'(x) = \frac{(x-4)^2 \cdot (x+2) \cdot 12x^2 - 4x^3\cdot((x-4)^2+(x+2)\cdot 2 \cdot (x-4))}{\left( (x-4)^2 \cdot (x+2)\right)^2} = \frac{-24x^2(x+4)}{(x-4)^3 \cdot (x+2)^2}

    f'(x) = 0 if the numerator equals zero. Thus you get 2 stationary points at x = 0 or x = -4. Minimum at x = -4: f(-4) = 2

    5. Points of inflection:

    Calculate the 2nd drivative of f. (You'll need at least half an hour ...)

    f''(x) = \frac{48x(x^3+8x^2+16x+32)}{(x-4)^4 \cdot (x+2)^3}

    f''(x) = 0 if the numerator equals zero. First you get x = 0 that means at x = 0 is no stationary point but a point of inflection with a horizontal slope. (Such a point is called in German a terrace point).

    The 2nd solution you'll get if
    x^3+8x^2+16x+32 = 0 ~\Longrightarrow~x\approx -6.26079... (I used a computer to get this solution)

    6. Graph (see attachment): Be careful the axes are scaled differently!
    Attached Thumbnails Attached Thumbnails Here's another (harder) one-gebr_rat_fkt3.gif  
    Last edited by earboth; August 14th 2007 at 11:15 AM.
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  3. #3
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    Thanx, Gr8 help
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