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Sketch a possible graph for a function that has the following characteristics. f(0)=2.5

-horizontal asymptote is y=3
-Vertical x=4
-f'(x)<0 and f''(x)>0 for x>4.
-f'(x)<0 and f''(x) <0 for x<4

I got the horizontal and vertical but I don't know what to do for the last two.

  • Calculus -

    the asymptotes make

    y = 3x/(x-4)

    a good starting place. We want f to be falling on both sides of the asymptote, kind of like y=1/x.

    y' = -12/(x-4)^2
    y' < 0 on both sides of x=4

    y'' = 24/(x-4)^3
    y'' < 0 for x<4
    y'' > 0 for x>4

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