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Question Number 143814 by mathdanisur last updated on 18/Jun/21

$$\mathrm{\forall }a;b;c\in \mathbb{R},findallf:\mathbb{R}\to \mathbb{R},suchthat$$$$f\left(a\right)f\left(bc\right)+9⩽f\left(ab\right)+5f\left(ac\right)$$

Answered by Olaf_Thorendsen last updated on 18/Jun/21

$$f\left(a\right)f\left(bc\right)+9⩽f\left(ab\right)+5f\left(ac\right)$$$$$$$$\mathrm{If}a=b=c=0:$$$$f^2\left(0\right)+9⩽6f\left(0\right)$$$$f^2\left(0\right)-6f\left(0\right)+9⩽0$$$${(f\left(0\right)-3)}^2⩽0\Rightarrow f\left(0\right)=3$$$$$$$$\mathrm{If}a=b=c=1:$$$$f\left(1\right)=3$$$$$$$$\mathrm{If}a=0\mathrm{and}\mathrm{c}=1:$$$$f\left(0\right)f\left(b\right)+9⩽f\left(0\right)+5f\left(0\right)$$$$3f\left(b\right)+9⩽6f\left(0\right)=18$$$$f\left(b\right)⩽3\mathrm{\forall }b\left(1\right)$$$$$$$$\mathrm{If}a=1\mathrm{and}b=0:$$$$f\left(1\right)f\left(0\right)+9⩽f\left(0\right)+5f\left(c\right)$$$$15⩽5f\left(c\right)$$$$f\left(c\right)⩾3\mathrm{\forall }c\left(2\right)$$$$$$$$\left(1\right)\mathrm{and}\left(2\right)\mathrm{\forall }x,f\left(x\right)=3$$

Commented by mathdanisur last updated on 20/Jun/21

$$thankyousircool$$

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