Let f ( x ) be a polynomial function with non-negative coefficients such that f...

Brunton39

Brunton39

Answered

2022-06-29

Let f ( x ) be a polynomial function with non-negative coefficients such that f ( 1 ) = f ( 1 ) = f ( 1 ) = f ( 1 ) = 1. Find the minimum value of f ( 0 )

Answer & Explanation

scoseBexgofvc

scoseBexgofvc

Expert

2022-06-30Added 20 answers

It is perhaps simpler to use Taylor's formula with fixed degree three and a Lagrange remainder:
f ( x ) = 1 + ( x 1 ) + 1 2 ! ( x 1 ) 2 + 1 3 ! ( x 1 ) 3 + f ( 4 ) ( ξ ) 4 ! ( x 1 ) 4 .
where ξ is between 1 and x. For x = 0 is ξ 0 and the last term is non-negative, since f ( 4 ) has non-negative coefficients as well. This gives
f ( 0 ) 1 1 + 1 2 1 6 = 1 3 .
The bound is sharp, equality holds for the function
f ( x ) = 1 + ( x 1 ) + 1 2 ! ( x 1 ) 2 + 1 3 ! ( x 1 ) 3 = 1 3 + 1 2 x + 1 6 x 3 .

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