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I need help proving that

\((c^{2}e^{2}+d^{2}f^{2}) \leq (c^{2}+d^{2})(e^{2}+f^{2})\)

I have tryed cauchy and got 

\((c^{2}e^{2}+d^{2}f^{2}) \leq \sqrt{(c^{4}+d^{4})(e^{4}+f^{4})}\)

 Oct 21, 2021
 #2
avatar+115340 
+2

Prove:

\((c^{2}e^{2}+d^{2}f^{2}) \leq (c^{2}+d^{2})(e^{2}+f^{2})\)

 

I have assumed that all pronumerals represent real numbers.

 

\(LHS=(c^{2}e^{2}+d^{2}f^{2}) \\ RHS= (c^{2}+d^{2})(e^{2}+f^{2})\\ RHS= c^{2}e^2+d^{2}f^2+c^2f^2+d^2f^2\\ RHS= LHS +c^2f^2+d^2f^2\\ since\;\;c^2\ge0,\quad f^2\ge0,\quad d^2\ge0 \quad and\quad f^2\ge0 \\ RHS= LHS +k\qquad \text{Where k is a real number }\ge0\\ LHS +k =RHS\\ LHS=RHS-k\\ LHS\le RHS\qquad QED \)

 Oct 21, 2021
 #3
avatar+12 
+1

wow, thanks!! and yes all the variables are real numbers

helpp  Oct 21, 2021

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