{2^{sin theta }} + {2^{cos theta }} is greate | vedclass

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(d) $\frac{1}{2}\left[ {{2^{\sin 	heta }} + {2^{\cos 	heta }}} ight] \ge \sqrt {{2^{\sin 	heta }}{2^{\cos 	heta }}} $

(${m{A}}{m{.M}}{m

Vedclass · Accepted Answer

${2^{\left( {1 - \,\frac{1}{{\sqrt 2 }}} \right)}}$

Vedclass · Answer

(d) $\frac{1}{2}\left[ {{2^{\sin 	heta }} + {2^{\cos 	heta }}} ight] \ge \sqrt {{2^{\sin 	heta }}{2^{\cos 	heta }}} $

(${m{A}}{m{.M}}{m{.}} \ge {m{G}}{m{.M}}{m{.}}$)

==> ${2^{\sin 	heta }} + {2^{\cos 	heta }} \ge {2.2^{(\sin 	heta + \cos 	heta )/2}}$ .....(i)

Now $(\sin 	heta + \cos 	heta ) = \sqrt 2 \sin (	heta + \pi /4) \ge - \sqrt 2 $

For all real $	heta$,

${2^{\sin 	heta }} + {2^{\cos 	heta }} \ge {2.2^{(\sin 	heta + \cos 	heta )/2}} > 2\,.\,{2^{ - \sqrt 2 /2}}$

==> ${2^{\sin 	heta }} + {2^{\cos 	heta }} \ge {2^{1 - (1/\sqrt 2 )}}$.

${2^{\sin \theta }} + {2^{\cos \theta }}$ is greater than

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