Find the Minimum value of 8 tan 2x + 7 cot 2x1). 3√562). 2√563)

1.	Find the Minimum value of 8 tan 2x + 7 cot 2x1). 3√562). 2√563). √564). 8√28
Answer» We have, $(8\;{\bf{ta}}{{\bf{N}}^2}{\bf{x}}\; + \;7{\cot ^2}{\bf{x}}\; = \;{\left( {\sqrt 8 \tan {\bf{x}}} \right)^2}\; + \;{\left( {\sqrt 7 \cot {\bf{x}}} \right)^2} - 2\; \times \;\sqrt 8 \tan {\bf{x}}\; \times \;\sqrt 7 \cot {\bf{x}}\; + \;2\; \times \;\sqrt 8 \tan {\bf{x}}\; \times \;\sqrt 7 \cot {\bf{x}})$ $(\Rightarrow \;{\left( {\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}} \right)^2}\; + \;2\sqrt {8\; \times \;7} \; = \;{\left( {\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}} \right)^2}\; + \;2\sqrt {56} )$ For the value to be minimum, $(\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}\; = \;0)$ ⇒ Minimum value = 2√56

Find the Minimum value of 8 tan 2x + 7 cot 2x1). 3√562). 2√563). √564). 8√28

Answer»

We have, $(8\;{\bf{ta}}{{\bf{N}}^2}{\bf{x}}\; + \;7{\cot ^2}{\bf{x}}\; = \;{\left( {\sqrt 8 \tan {\bf{x}}} \right)^2}\; + \;{\left( {\sqrt 7 \cot {\bf{x}}} \right)^2} - 2\; \times \;\sqrt 8 \tan {\bf{x}}\; \times \;\sqrt 7 \cot {\bf{x}}\; + \;2\; \times \;\sqrt 8 \tan {\bf{x}}\; \times \;\sqrt 7 \cot {\bf{x}})$

$(\Rightarrow \;{\left( {\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}} \right)^2}\; + \;2\sqrt {8\; \times \;7} \; = \;{\left( {\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}} \right)^2}\; + \;2\sqrt {56} )$

For the value to be minimum, $(\sqrt 8 \tan {\bf{x}} - \sqrt 7 \cot {\bf{x}}\; = \;0)$

⇒ Minimum value = 2√56

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