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  4. If f (θ)=2( sec2 θ+ cos2 θ), then its value always

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Q. If $f \left(\theta\right)=2\left(\sec^{2}\,\theta+ \cos^{2}\,\theta\right)$, then its value always

UPSEEUPSEE 2016

Solution:

Given, $f(\theta) =2\left(\sec ^{2} \theta+\cos ^{2} \theta\right)$ ...(i)
Let $a =2 \sec ^{2} \theta$ and $b=2 \cos ^{2} \theta$
We know that, $\frac{a +b}{2} \geq \sqrt{a b}$
$2 \sec ^{2} \theta+2 \cos ^{2} \theta \geq 2 \sqrt{2 \sec ^{2} \theta \cdot 2 \cos ^{2} \theta}$
$\Rightarrow 2\left(\sec ^{2} \theta+\cos ^{2} \theta\right) \geq 2 \sqrt{4}$
$\Rightarrow 2\left(\sec ^{2} \theta+\cos ^{2} \theta\right) \geq 4$
$f(\theta) \geq 4$ [from Eq (i)]