Minimum of maximum

Algebra Level 5

Consider all monic polynomials of the form fa(x)=x+af_a(x) = x + a . As aa ranges over all real numbers, what is the minimum value of NaN_a, where

Na=maxx[16,10]fa(x)? N_a = \max_{x \in [-16, 10] } \lvert f_a(x) \rvert ?

Details and assumptions

The last equation states: "The maximum value of the absolute value of fa(x)f_a(x), as xx ranges from 16-16 to 1010 inclusive".

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