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If α 2+β 2+...

Question

If $α^{2} + β^{2} + γ^{2} = 1$ , then highest integral value of $α β + β γ + α γ$ is

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Solution

$\begin{matrix} α^{2} + β^{2} + γ^{2} = 1 {(α - β)}^{2} {(β - γ)}^{2} + {(γ - α)}^{2} \geq 0 2 [α^{2} + β^{2} - α β - β γ - γ α] \geq 0 1 - (α β + β γ + γ α) \geq 0 α β + β γ + γ α - 1 \leq 0 α β + β γ + γ α 1 ∴ H e i g h t i n t e g r a l v a l u e i s 1 \end{matrix}$

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