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a-b-c-R-find-all-f-R-R-such-that-f-a-f-bc-9-f-ab-5f-ac-

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Question Number 143814 by mathdanisur last updated on 18/Jun/21
∀a;b;c∈R , find all f:R→R , such that f(a)f(bc)+9≤f(ab)+5f(ac)
∀a;b;c∈R,findallf:R→R,suchthatf(a)f(bc)+9⩽f(ab)+5f(ac)
Answered by Olaf_Thorendsen last updated on 18/Jun/21
f(a)f(bc)+9 ≤ f(ab)+5f(ac) If a = b = c = 0 : f^2 (0)+9 ≤ 6f(0) f^2 (0)−6f(0)+9 ≤ 0 (f(0)−3)^2 ≤ 0 ⇒ f(0) = 3 If a = b = c = 1 : f(1) = 3 If a = 0 and c= 1 : f(0)f(b)+9 ≤ f(0)+5f(0) 3f(b)+9 ≤ 6f(0) = 18 f(b) ≤ 3 ∀b (1) If a = 1 and b = 0 : f(1)f(0)+9 ≤ f(0)+5f(c) 15 ≤ 5f(c) f(c) ≥ 3 ∀c (2) (1) and (2) ∀x, f(x) = 3
f(a)f(bc)+9⩽f(ab)+5f(ac)Ifa=b=c=0:f2(0)+9⩽6f(0)f2(0)−6f(0)+9⩽0(f(0)−3)2⩽0⇒f(0)=3Ifa=b=c=1:f(1)=3Ifa=0andc=1:f(0)f(b)+9⩽f(0)+5f(0)3f(b)+9⩽6f(0)=18f(b)⩽3∀b(1)Ifa=1andb=0:f(1)f(0)+9⩽f(0)+5f(c)15⩽5f(c)f(c)⩾3∀c(2)(1)and(2)∀x,f(x)=3
Commented by mathdanisur last updated on 20/Jun/21
thankyou sir cool
thankyousircool

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