Inequality Proof: Max {A+B,C} ≤ Max {A,C} + Max {B,C}

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SUMMARY

The inequality \(\max\{A+B,C\} \leq \max\{A,C\} + \max\{B,C\}\) holds true when A, B, and C are nonnegative real numbers. However, counterexamples exist when negative values are introduced, demonstrating that the inequality does not universally apply. The discussion emphasizes the importance of analyzing specific cases, particularly when \(\max\{A+B,C\} = C\) or \(\max\{A+B,C\} = A+B\), to identify potential violations of the inequality.

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Hello,

i've met during problem solving with inequality

\max\{A+B,C\}\le\max\{A,C\}+\max\{B,C\}

where A,B and C are real numbers. I don't know whether it holds, but I need to prove that.

Thanks for reply...
 
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It holds if A,B,C are nonnegative, but play around with some negative numbers and you'll find a counterexample. (Hint: Does the inequality always hold if max{A+B,C}=A+B, or if max{A+B,C}=C? Pick the case you can't prove easily and search for counterexamples there.)
 

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