If x, y, z are positive real numbers then the minimum value of x4+y4+z2xyz is k1 and minimum value of sec4⁡αtan2⁡β+sec4⁡βtan2⁡α,α,β∈0,π2 is k2 The sum of digits ofk12+k22 equals to

Using A.M. ≥ G.M. for x4,y4,z22,z22 , we get k1=22using a2+b2≥2ab we get,Given expression ≥2sec2⁡αsec2⁡βtan⁡βtan⁡α≥8sin⁡2αsin⁡2β≥8⇒ k12+k22=8+64=72Sum of digits = 9

If x, y, z are positive real numbers then the minimum value of x4+y4+z2xyz is k1 and minimum value of sec4⁡αtan2⁡β+sec4⁡βtan2⁡α,α,β∈0,π2 is k2 The sum of digits ofk12+k22 equals to

Using A.M. ≥ G.M. for x4,y4,z22,z22 , we get k1=22using a2+b2≥2ab we get,Given expression ≥2sec2⁡αsec2⁡βtan⁡βtan⁡α≥8sin⁡2αsin⁡2β≥8⇒ k12+k22=8+64=72Sum of digits = 9

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If x, y, z are positive real numbers then the minimum value of $\frac{x^{4} + y^{4} + z^{2}}{x y z}$ is k₁ and minimum value of $\frac{\sec^{4} α}{\tan^{2} β} + \frac{\sec^{4} β}{\tan^{2} α}, α, β \in (0, \frac{π}{2})$ is k₂ The sum of digits of
$k_{1}^{2} + k_{2}^{2}$ equals to

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Detailed Solution

Using A.M. $\geq$ G.M. for $x^{4}, y^{4}, \frac{z^{2}}{2}, \frac{z^{2}}{2}$ , we get $k_{1} = 2 \sqrt{2}$
using $a^{2} + b^{2} \geq 2 a b$ we get,
Given expression $\geq \frac{2 \sec^{2} {αsec}^{2} β}{\tan βtan α} \geq \frac{8}{\sin 2 αsin 2 β} \geq 8$
$\Rightarrow k_{1}^{2} + k_{2}^{2} = 8 + 64 = 72$
Sum of digits = 9