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Thread: For what values of k will the curve have exactly one tangent line?

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    For what values of k will the curve have exactly one tangent line?

    This is my first post so I hope I'm doing this correctly...
    Q: For what value(s) of the constant will the curve y=x^3+kx^2+3x-4 have exactly one horizontal tangent?

    I know:
    Step 1: take a derivative:

    y'=3x^2+2kx+3

    Step 2: set it equal to 0 and solve for x

    This is where I get stuck. This does not factor nicely, so we have to use the quadratic formula. I got a little help from a classmate, but I'm not understanding the "WHY" part. Here is what I have, we have to use just the discriminant part (b^2-4ac) to find our answer. I know the answer is k=-3, 3 and I can complete the math part, it is just the understanding part that I'm lost on. Why wouldn't we use the whole quadratic formula? How is it legal to just use part of it? I've attached a picture of the work I completed in class. Thanks in advance for any help you all can give!!

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  2. #2
    Elite Member stapel's Avatar
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    Cool

    Quote Originally Posted by bluelucky7 View Post
    Q: For what value(s) of the constant will the curve y = x^3 + kx^2 + 3x - 4 have exactly one horizontal tangent?

    I know:
    Step 1: take a derivative:

    y'=3x^2+2kx+3

    Step 2: set it equal to 0 and solve for x

    This is where I get stuck. This does not factor nicely, so we have to use the quadratic formula. I got a little help from a classmate, but I'm not understanding the "WHY" part. Here is what I have, we have to use just the discriminant part (b^2-4ac) to find our answer. I know the answer is k=-3, 3 and I can complete the math part, it is just the understanding part that I'm lost on. Why wouldn't we use the whole quadratic formula? How is it legal to just use part of it?
    Think back to what you learned when you first worked with the Quadratic Formula? What did the discriminant (the part inside the square root) tell you about the real-valued solutions (that is, the x-intercepts) of the graph of the related quadratic function?

    If the discriminant is negative, are there any graphable solutions? In this context, are there any solutions that give you a graphable horizontal tangent?

    If the discriminant is positive, how many solutions are there (from the "plus / minus" in front of the radical)? In this context, how many graphable horizontal tangent points do you get?

    On the other hand, if the discriminant is exactly zero, where does this lead?

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