Question

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

• A. $\ge 4$
• B. $>4$
• C. $<4$
• D. None of these

Answer 1

Answer: (A)

We have, $4^{{\sin }^{2}x}+4^{{\cos }^{2}x}\ge 2\sqrt{4^{{\sin }^{2}x}\cdot 4^{{\cos }^{2}x}}$
$[\because AM\ge GM]$
$\Rightarrow 4^{{\sin }^{2}x}+4^{{\cos }^{2}x}\ge 2\sqrt{4^{\left({\sin }^{2}x+{\cos }^{2}x\right)}}$
$\Rightarrow 4^{{\sin }^{2}x}+4^{{\cos }^{2}x}\ge 2\sqrt{4}$
$\Rightarrow 4^{{\sin }^{2}x}+4^{{\cos }^{2}x}\ge 4$