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The purpose of this letter is to investigate the time complexity consequencesof the truncated Taylor series, known as Taylor Polynomials\cite{bakas2019taylor,Katsoprinakis2011,Nestoridis2011}. In particular, it isdemonstrated that the examination of the $\mathbf{P=NP}$ equality, isassociated with the determination of whether the $n^{th}$ derivative of aparticular solution is bounded or not. Accordingly, in some cases, this is nottrue, and hence in general.

The Lagrangian remainder of Taylor's series, distinguishes $\mathcal{O}(f(x))$ time complexities to polynomials or not