The number of different possible values for t | vedclass

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Question

(c)

We have,

$x^4+4 y^4+16 z^4+64=32 x y z$

We know $\quad AM \geq GM$

$\Rightarrow \frac{x^4+4 y^4+16 z^4+64}{4}$

$\geq\left(x^4 	ime

Vedclass · Accepted Answer

$4$

Vedclass · Answer

(c)

We have,

$x^4+4 y^4+16 z^4+64=32 x y z$

We know $\quad AM \geq GM$

$\Rightarrow \frac{x^4+4 y^4+16 z^4+64}{4}$

$\geq\left(x^4 	imes 4 y^4 	imes 16 z^4 	imes 64ight)^{1 / 4}$

$\Rightarrow \quad x^4+4 y^4+16 z^4+64 \geq 32 x y z$

$\Rightarrow \quad x=\pm 2 \sqrt{2}, y=\pm 2, z=\pm \sqrt{2}$

For $x, y, z$

For $x^4+4 y^4+16 z^4+64=32 x y z$

Either each of $x, y, z$ is positive $ightarrow 1$ case

or two of $x, y, z$ are negative $ightarrow 3$ cases

$	herefore 4$ cases of different $(x, y, z)$ triplets

$4$ possible $x+y+z$ values $(x 
eq y 
eq z)$.

The number of different possible values for the sum $x+y+z$, where $x, y, z$ are real number such that $x^4+4 y^4+16 z^4+64=32 x y z$ is

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