If y = y(x), then the derivative with respect to x is dy/dx. Differentiating both sides of the given equation gives
d/dx [4x ² + y ²] = d/dx [36]
8x + 2y dy/dx = 0
2y dy/dx = -8x
dy/dx = -4x/y
The turning points of the curve, taken as a function of x, are those points where the derivative vanishes.
-4x/y = 0 ===> x = 0
This value of x corresponds to two points on the curve,
4×0² + y ² = 36 ===> y ² = 36 ===> y = ±6
So there are two turning points, (0, -6) and (0, 6).
