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Determine-all-function-f-R-R-which-satisfy-f-a-x-f-a-x-4ax-for-all-real-a-and-x-




Question Number 120300 by bemath last updated on 30/Oct/20
Determine all function f:R→R  which satisfy f(a+x)−f(a−x)=4ax  for all real a and x.
Determineallfunctionf:RRwhichsatisfyf(a+x)f(ax)=4axforallrealaandx.
Commented by benjo_mathlover last updated on 30/Oct/20
let f(x)=px^2 +qx+r  f(a+x)=p(a^2 +2ax+x^2 )+qx+aq+r  f(a+x)=px^2 +(2ap+q)x+a^2 p+aq+r    f(a−x)=p(a^2 −2ax+x^2 )+aq−qx+r  f(a−x)=px^2 −(2ap+q)x+a^2 p+aq+r    f(a+x)−f(a−x)=(4ap+2q)x ≡ 4ax  → 4ap+2q = 4a ; 4ap−4a=−2q  4a(1−p)= 2q ⇒ 1−p=(q/(2a))  ⇒p=1−(q/(2a)) = ((2a−q)/(2a))  f(x)= (((2a−q)/(2a)))x^2 +qx+r
letf(x)=px2+qx+rf(a+x)=p(a2+2ax+x2)+qx+aq+rf(a+x)=px2+(2ap+q)x+a2p+aq+rf(ax)=p(a22ax+x2)+aqqx+rf(ax)=px2(2ap+q)x+a2p+aq+rf(a+x)f(ax)=(4ap+2q)x4ax4ap+2q=4a;4ap4a=2q4a(1p)=2q1p=q2ap=1q2a=2aq2af(x)=(2aq2a)x2+qx+r

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