This chapter is an elementary introduction in Optimization Theory. See an excellent book Brinkus J. (and Tikhomirov V. 2005) for other details. Optimization of functions in two variables. Bolzano-Weierstrass Lemma and Weierstrass’ Theorem. The critical points theorem. Local minima by gradients and level curves. The method of the steepest descent. Clairaut’s Theorem. The method of Hessian matrices. Classification of critical points for classical curves by Descartes, Bernoulli, and Cramer. Constraint optimization. The method of Lagrange’s multipliers.