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In this paper, we first study the problem of finding the maximum number of zeros of functions with parameters and then apply the results obtained to smooth or piecewise smooth planar autonomous systems and scalar periodic equations to study the number of limit cycles or periodic solutions, improving some fundamental results both on the maximum number of limit cycles bifurcating from an elementary focus of order $k$ or a limit cycle of multiplicity $k$, or from a period annulus, and on the maximum number of periodic solutions for scalar periodic smooth or piecewise smooth equations as well.