Tangencies for Power Functions with Integer Exponent - Vaz Cotrim, Cairo Henrique (2025)
Considering a power function f(x) = x^n with exponent n as a positive integer, we show that, at each of its points, there exists a unique polynomial function of degree n − 1 that is tangent to it at that point. Similarly, we verify that every power function h(x) = x^k with exponent k as a negative integer is tangent, at each of its points, to a function of the form l(x) =Sa^t.x^t, where the exponents t are integers between k + 1 and −1.