Horizontal tangent line to a cubic:

Where does  have a tangent line parallel to the x axis?

Since the x axis, a horizontal line, has a slope of 0, the desired parallel line tangent to the given curve, will also have a slope of 0. This slope is also equal to the value of the derivative of y at the point of tangency (x_T, y_T). Hence, at this point we have

, or at (x_T, y_T), 

Now we can substitute these values of x into the original equation to find the corresponding y coordinates:

, and

Hence the points of tangency are and  . See the approximate graph.

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