In Exercise 43, implicit differentiation was used to find the relative extrema. The exercise was contrived to avoid various difficulties that could have arisen. Discuss some of the difficulties that might be encountered in such problems, and how these difficulties might be resolved.

Exercise 43

Suppose x and y are related by the equation -12x + x³ + y + y² = 4

(a) Find all critical points on the curve.
(b) Determine whether the critical points found in part (a) are relative maxima or relative minima by taking values of x nearby and solving for the corresponding values of y.