Prove f(x) is quadratic if f ( 2 x ) = 4 f ( x ) and f(x) is increasing over positive xThe problem arose in the context of kinetic energy, where it can be proven from symmetry principles that E ( 2 v ) = 4 E ( v ) without assuming E = m v 2 / 2While one may do further physics from this point to prove the desired result (that E is quadratic in v) -- consider a system with other prime numbers of balls, then do algebra to prove the result for rational scaling in v, then use the fact that there are rational numbers between any two real numbers and assume the function is increasing to prove it for all real scaling -- it seems intuitively obvious from this point immediately, that if E is increasing in v, E = k m v 2 .How would one prove this functional equation?

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Prove f(x) is quadratic if   f  (  2  x  )  =  4  f  (  x  ) and f(x) is increasing over positive xThe problem arose in the context of kinetic energy, where it can be proven from symmetry principles that   E  (  2  v  )  =  4  E  (  v  ) without assuming   E  =  m      v    2        /    2While one may do further physics from this point to prove the desired result (that E is quadratic in v) -- consider a system with other prime numbers of balls, then do algebra to prove the result for rational scaling in v, then use the fact that there are rational numbers between any two real numbers and assume the function is increasing to prove it for all real scaling -- it seems intuitively obvious from this point immediately, that if E is increasing in v,   E  =  k  m      v    2  .How would one prove this functional equation?

cdtortosadn · Accepted Answer

Step 1If we assume   f  ∈      C    2   (or at least that f is twice differentiable and f′′ is continuous at 0), we can prove uniqueness. Taking the derivative twice, we obtain:  f  (  2  x  )  =  4  f  (  x  )    ⟹        f    ″    (  2  x  )  =      f    ″    (  x  )Step 2Now we have, using continuity of f′′ at 0:      f    ″    (  x  )  =      f    ″        (          x      2        )    =      f    ″        (          x      4        )    =  …  =      f    ″        (          x              2        n              )        →          n      →      ∞            f    ″    (  0  )Hence, f′′ is constant so   f  (  x  )  =  a      x    2    +  b  x  +  c  f  (  x  )  =  a      x    2    ,    a  &amp;gt;  0

cousinhaui · Accepted Answer

Step 1This is false.Take an arbitrary increasing function f defined on [1,2] taking values in [1,4].Then define   f  (  x  )  =      4          −      n        f  (      2    n    x  ) whenever   x  ∈  [      2          −      n        ,      2          1      −      n        ]. Then f is increasing and   f  (  2  x  )  =  4  f  (  x  ), but f is not in general quadratic.Step 2Moreover, if we define   f  (  0  )  =  0, then f is continuous at 0, since   f  (  x  )  ≤      4          −      n        f  (  2  ) whenever   x  ∈  [      2          −      n        ,      2          1      −      n        ].

Prove f(x) is quadratic if f(2x)=4f(x) and f(x) is increasing over positive x

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