Optimization is introduced to teaching in secondary school, focussing on functions of one variable with the first derivative criterion for critical points and monotonicity of functions. The second derivative criterion is also used to determine the nature of the critical points. Similarly, in university-level teaching, algebraic work using criteria is the predominant approach adopted for optimization of two-variable functions, but with much less meaning than for one-variable functions. To recover meaning, we designed a learning situation based on local approximations with the first- and second-order Taylor polynomial, both for one- and two-variable functions. The results show that the local visualization for first-order Taylor approximation was quite easy for students to understand (curve and tangent), but local visualization for second-order Taylor approximation (curve and osculatory parabola) was much less so. We also identify a competition between local and global visualization processes, which constitutes a newly identified visualization phenomenon.