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Math Help - last function problem

  1. #1
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    last function problem

    Show that the graph f(x)=(x^3)+(2x^2)+6x does not have a tangent line with a slope of 4.

    Im not really sure where to start with this one...find f'(x) first or what?
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  2. #2
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    Quote Originally Posted by whoislambo View Post
    Show that the graph f(x)=(x^3)+(2x^2)+6x does not have a tangent line with a slope of 4.

    Im not really sure where to start with this one...find f'(x) first or what?
    Hi

    Yes , you differentiate first to find the gradient of the tangent line .

    f'(x)=3x^2+4x+6

    Now check its discriminant

    b^2-4ac=4^2-4(3)(6)=-56<0

    Since its smaller than 0 , hence it has no real roots , which means no value of x would give you 4 , which is the gradient of the tangent line .
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  3. #3
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by whoislambo View Post
    Show that the graph f(x)=(x^3)+(2x^2)+6x does not have a tangent line with a slope of 4.

    Im not really sure where to start with this one...find f'(x) first or what?
    Find f'(x) and show it can never equal 4.

    f'(x)=3x^2+4x+6

    Show 3x^2+4x+6=4 has no real solutions.

    Spoiler:
    Use the quadratic formula on 3x^2+4x+2=0, so a=3, b=4, and c=2.

    \frac{-4\pm\sqrt{16-24}}{6}=\frac{-4\pm\sqrt{-8}}{6}=-\frac{2}{3}\pm \frac{\sqrt{2}}{3}i

    So there are no real roots, and therefore the slope of the tangent line never equals 4. (You actually could have stopped as soon as you saw that the discriminant in the quadratic formula was negative, but I figured I'd finish it out to the end.)
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