William W. answered • 10/05/20

Experienced Tutor and Retired Engineer

Use implicit differentiation like this:

To find the points where the graph has a horizontal tangent set dy/dx equal to zero. Note that the ONLY way a rational expression can equal zero is if the numerator equals zero. So 2x = 0 or x = 0. Then since y^{4} = y^{2} - x^{2}, then:

y^{4} = y^{2} - 0

y^{4} - y^{2} = 0

y^{2}(y^{2} - 1) = 0

y^{2}(y + 1)(y - 1) = 0

y = 0, y = -1, y = 1 but notice that y = 0 will give you a derivative with the denominator equal to zero so that cannot be a solution. That means (0, 1) and (0, -1) have horizontal tangent lines.

Vertical tangent lines will occur when the derivative is undefined. To find those point, set the denominator equal to zero.

Do the same process on the other equation