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Math Help - Points of Symmetry and Turning Points

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    Points of Symmetry and Turning Points

    The formula for findintg the x-coordinate of the point of symmetry of a cubic function is -b/3a. Use this formula to find the point of symmetry (x and y axes) of the following function:

    ax^3+bx^2+d

    Also find the two turning points on this function.

    Show that the point of symmetry is the midpoint of the line segment connecting the local maximum and local minimum.

    PLEASE SHOW ALL STEPS
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    Quote Originally Posted by ineedfriendlymathhelp View Post
    The formula for findintg the x-coordinate of the point of symmetry of a cubic function is -b/3a. Use this formula to find the point of symmetry (x and y axes) of the following function:

    ax^3+bx^2+d

    Also find the two turning points on this function.

    Show that the point of symmetry is the midpoint of the line segment connecting the local maximum and local minimum.

    PLEASE SHOW ALL STEPS
    With a function of degree 3 the point of symmetry is the same as the point of inflection. You'll get the x-coordinate of this point by setting the second derivation equal to zero:

    f(x)=ax^3+bx^2+d~\implies~f''(x)=6ax +3b

    6ax+2b=0~\implies~x=-\dfrac b{3a} and therefore the point of symmetry is S\left(-\dfrac b{3a} , \dfrac {2b^3}{27a^2} +d\right)

    The condition for a turning point is f'(x) = 0

    With your function you'll get:

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

    3ax^2+2bx=0~\implies~x(3ax+2b)=0~\implies~x=0~\vee  ~x=-\dfrac{2b}{3a}

    Therefore T_1\left(0,d\right) and T_2\left(-\dfrac{2b}{3a}, \dfrac{4b^3}{27a^2}+d \right)

    Now calculate the mean values of the coordinates of T_1 and T_2 and you'll get the coordinates of point S.
    Last edited by earboth; November 3rd 2008 at 04:20 AM. Reason: corrected a minor mistake
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