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  4. For all real x, 4 sin2x+ 4 cos2x is

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Q. For all real $ x, 4^{\sin^2x}+ 4\cos^{2x} $ is

AMUAMU 2015

Solution:

We have, $4^{\sin ^{2} x}+4^{\cos ^{2} x} \geq 2 \sqrt{4^{\sin ^{2} x} \cdot 4^{\cos ^{2} x}}$
$[\because A M \geq G M]$
$\Rightarrow 4^{\sin ^{2} x}+4^{\cos ^{2} x} \geq 2 \sqrt{4^{\left(\sin ^{2} x+\cos ^{2} x\right)}}$
$\Rightarrow 4^{\sin ^{2} x}+4^{\cos ^{2} x} \geq 2 \sqrt{4}$
$\Rightarrow 4^{\sin ^{2} x}+4^{\cos ^{2} x} \geq 4$