Let a , b ∈ R with a&lt;b and f : [ a , b ] → [ a , b ] be a continuous function. If a 0 , a 1 , . . . , a n ∈ R and a 0 + a 1 + a 2 + . . . + a n ≠ 0 and a 0 x + a 1 f ( x ) + a 2 f 2 ( x ) + . . . + a n f n ( x ) = 0 , ∀ x ∈ [ a , b ], prove that a ⋅ b &lt; 0

Question

Let   a  ,  b  ∈      R   with a&amp;lt;b and   f  :  [  a  ,  b  ]  →  [  a  ,  b  ] be a continuous function. If       a    0    ,      a    1    ,  .  .  .  ,      a    n    ∈      R   and       a    0    +      a    1    +      a    2    +  .  .  .  +      a    n    ≠  0 and       a    0    x  +      a    1    f  (  x  )  +      a    2        f    2    (  x  )  +  .  .  .  +      a    n        f    n    (  x  )  =  0  ,  ∀  x  ∈  [  a  ,  b  ], prove that   a  ⋅  b  &amp;lt;  0

imperiablogyy · Accepted Answer

The statement is not true.Consider   f  :  [  1  ,  4  ]  →  [  1  ,  4  ] defined for all   x  ∈  [  1  ,  4  ] by  f  (  x  )  =  2      x  (so a=1 and b=4).Let       a    0    =  −  4,       a    1    =  0 and       a    2    =  1. Then one has      a    0    +      a    1    +      a    2    ≠  0and for all   x  ∈  [  0  ,  1  ]      a    0    x  +      a    1    f  (  x  )  +      a    2        f    2    (  x  )  =  −  4  x  +  4  x  =  0but   a  b  &amp;gt;  0

If a_0+a_1+...+a_n != 0 and a_0 x+a_1 f(x)+a_2 f^2(x)+...+a_n f^n(x)=0 AA x in [a,b], prove that a * b<0

Answered question

Answer & Explanation

New Questions in Algebra I