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Math Help - Horizontal line

  1. #1
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    Horizontal line

    Let f(x)=ax^3+bx^2+cx+d , find conditions for a,b.c and d so that

    a) f has no horizontal tangent line.
    b) f has aunique horizontal tangent line.
    c) f has tow horizontal tangent lines
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  2. #2
    Behold, the power of SARDINES!
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    Quote Originally Posted by BAHADEEN View Post
    Let f(x)=ax^3+bx^2+cx+d , find conditions for a,b.c and d so that

    a) f has no horizontal tangent line.
    b) f has aunique horizontal tangent line.
    c) f has tow horizontal tangent lines
    If you take the derivative you get

    f'(x)=3ax^2+2bx+c

    So a graph has a horizontal tangent when the derivative is equal to zero.

    so we need to analyze the discriminant of the quadratic.

    Can you finish from here?
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  3. #3
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    Quote Originally Posted by TheEmptySet View Post
    If you take the derivative you get

    f'(x)=3ax^2+2bx+c

    So a graph has a horizontal tangent when the derivative is equal to zero.

    so we need to analyze the discriminant of the quadratic.

    Can you finish from here?
    How could I find the three conditions? thank you for help
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  4. #4
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
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    Quote Originally Posted by BAHADEEN View Post
    How could I find the three conditions? thank you for help
    The discrimiant is

    (2b)^2-4(3a)(c)=4b^2-12ac=4(b^2-3ac)

    This will have no solutions when the discrimiant is negative.

    This will have exactly one solution when the discrimiant is equal to zero.

    This will have two solutions when the discrimiant is positive.
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