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Math Help - Tangent Lines Passing Through Origin

  1. #1
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    Tangent Lines Passing Through Origin

    At how many points on the curve y=4x^5-3x^4+15x^2+6 will the line tangent to the curve pass through the origin.

    The only way I know how to go about this problem is to find the derivative and thats it. Please Help!
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  2. #2
    Senior Member Dinkydoe's Avatar
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    It depends on how many different roots x_0 \in \mathbb{R} the equation f(x) = 0 has.

    Observe that between any 2 different roots x_1,x_2 that f(x) attains a maximum/minimum in the interval (x_1,x_2).

    And there exists a a_0 with x_1<a_0<x_2 where f'(a_0)(x-a_0) +f(a_0) the tangent at (a_0,f(a_0)) passes through the origin.

    But we might just as well calculate how many maxima/minima f(x) attains:
    Thus if you find all different "real " roots of f'(x) = 20x^4-12x^3+30x = x(20x^3-12x^2+30) = 0 you're done.

    Edit: You don't even have to find the roots of f'(x)= 0 explicitly. You can use the intermediate value theorem to decide how many roots in \mathbb{R} this equation has.
    Last edited by Dinkydoe; January 17th 2010 at 10:10 AM.
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  3. #3
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    If you continue by using \frac{f(x)}{x}

    then you can find the minimum value(s) of this,
    that will give you the slope of the tangent(s) that passes through the origin,
    allowing for x being positive or negative.
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  4. #4
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    In blue is f(x) and in pink is \frac{f(x)}{x}

    the tangent slope gives the min value for \frac{f(x)}{x}

    since if another x is chosen to the right of the origin, the ratio will be greater.
    Attached Thumbnails Attached Thumbnails Tangent Lines Passing Through Origin-tangents.jpg  
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