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.

Question

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.

Ryan Newman · Accepted Answer

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    +  cAfter 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

Consider a , b , c ∈ <mi mathvariant="double-struck">R such that

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