Mathematics

If $\alpha ^{2}+\beta ^{2}+\gamma ^{2}=1$, then highest integral value of $\alpha \beta +\beta \gamma +\alpha \gamma$ is

SOLUTION
$\begin{array}{l} { \alpha ^{ 2 } }+{ \beta ^{ 2 } }+{ \gamma ^{ 2 } }=1 \\ { \left( { \alpha -\beta } \right) ^{ 2 } }{ \left( { \beta -\gamma } \right) ^{ 2 } }+{ \left( { \gamma -\alpha } \right) ^{ 2 } }\ge 0 \\ 2\left[ { { \alpha ^{ 2 } }+{ \beta ^{ 2 } }-\alpha \beta -\beta \gamma -\gamma \alpha } \right] \ge 0 \\ 1-\left( { \alpha \beta +\beta \gamma +\gamma \alpha } \right) \ge 0 \\ \alpha \beta +\beta \gamma +\gamma \alpha -1\le 0 \\ \alpha \beta +\beta \gamma +\gamma \alpha 1 \\ \therefore \, \, Height\, \, { { int } }egral\, \, value\, \, is\, \, 1 \end{array}$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

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