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crossoverman9b 2022-06-26 Answered
Consider a , b , c R such that | a x 2 + b x + c | 1 x [ 0 , 1 ] . Prove that | a | 8 , | b | 8 and | c | 1.
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Answers (1)

Ryan Newman
Answered 2022-06-27 Author has 26 answers
Let f ( x ) = a x 2 + b x + c.
Hence, f ( 0 ) = c, f ( 1 ) = a + b + c and f ( 1 2 ) = a 4 + b 2 + c
After solving of this system we obtain:
| a | = | 2 f ( 1 ) 4 f ( 1 2 ) + 2 f ( 0 ) | 2 | f ( 1 ) | + 4 | f ( 1 2 ) | + 2 | f ( 0 ) | 8
| b | = | f ( 1 ) + 4 f ( 1 2 ) 3 f ( 0 ) | | f ( 1 ) | + 4 | f ( 1 2 ) | + 3 | f ( 0 ) | 8
| c | = | f ( 0 ) | 1
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