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Math Help - tangent parallel to y or x axis

  1. #1
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    tangent parallel to y or x axis

    Find the critical points for each of the following. Determine whether the critical point is a local max or min and whether or not the tangent is parallel to the horizontal axis.

    y=(-x^2)(e^(-3x))

    so i had no problem finding the critical points and max or min points, but I am wondering about whether or not the tangent is parallel to the horizontal axis.
    In the back of the book the answer says:

    (0,0) is a local max, tangent parallel to the t-axis, (2/3, -4/(9e^2)) is a local min, tangent parallel to t-axis.

    Ok, I am thinking that t-axis is a typo and supposed to be y-axis for both cases. BUT, I am not sure why this is true (both parallel to the y-axis).

    When I graph it, they both look parallel to the x-axis to me. How do you know if it is parallel to y-axis?
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  2. #2
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    its suppose to be x-axis

    if you get any curve object you have lying around (and holding it in any position you like) and use a pencil as the tangent to the point closest to the ground (a local min), that tangent (the pencil) is always horizontal.
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  3. #3
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    Hi skeske1234

    You find the value of x = 0 and x = 2/3 by setting \frac{dy}{dx}=0.

    The slope of tangent is equal to \frac{dy}{dx}. For x = 0 and x = 2/3, of course \frac{dy}{dx}=0. It means that the slope of tangents at both value of x is zero.

    If the slope of a line is zero, will it parallel to x-axis or y-axis?
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  4. #4
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    Quote Originally Posted by songoku View Post
    Hi skeske1234

    You find the value of x = 0 and x = 2/3 by setting \frac{dy}{dx}=0.

    The slope of tangent is equal to \frac{dy}{dx}. For x = 0 and x = 2/3, of course \frac{dy}{dx}=0. It means that the slope of tangents at both value of x is zero.

    If the slope of a line is zero, will it parallel to x-axis or y-axis?
    Ok, but I thought about it and I think that the (0,0)'s tangent is supposed to be parallel to y-axis and (2/3, -4/(9e^(2))'s tangent is suppposed to be parallel to the x-axis. I think this is tre because when I do the f'(x) test, I note that from x<2/3 (including x<0), f'(x) is negative and x>2/3 is positive.. so, since x<0 and 0<x<2/3 is negative, that means parallel to the y axis or vertical tangent doesnt it?
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  5. #5
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    Hi skeske1234

    No, f '(x) test helps us to know about the nature of the function. For x < 0, f '(x) is positive. It means that for -\infty < x < 0, the function is increasing. For 0 < x < 2/3, f '(x) is negative, so the function on that interval is decreasing. For x > 2/3, f '(x) is positive so the function is increasing.

    The tangents at x = 0 and x = 2/3 have the same slope so the tangents will have same orientation. If the tangent at x = 2/3 is parallel to x-axis, which is true, the tangent at x = 2/3 will also parallel to x-axis.
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