Math Help - Points of Symmetry and Turning Points

1. 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.

2. Originally Posted by ineedfriendlymathhelp
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.

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 $f'(x) = 0$

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