If x-y-z -0-1 such that x - y - z - 1- then the least value of 1-x1-y1-zxyz is

Using A.M. ≥ G.M., we get x+y≥2√(xy), y+z≥2√(yz), z+x≥2√(zx) Multiplying all three inequalities, we get (x+y)(y+z)(z+x)≥ 8xyz Since x+y+z=1, ∴(1 z)(1 x)(1 y)≥8x ...

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Byju's Answer

Standard VII

Physics

Speed (Average Speed)

If x,y,z ∈0,1...

Question

If $x, y, z \in (0, 1)$ such that $x + y + z = 1,$ then the least value of $\frac{(1 - x) (1 - y) (1 - z)}{x y z}$ is

8

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4

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8 \sqrt{2}

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4 \sqrt{2}

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Solution

The correct option is A $8$
Using A.M. $\geq$ G.M., we get
$x + y \geq 2 \sqrt{x y},$
$y + z \geq 2 \sqrt{y z},$
$z + x \geq 2 \sqrt{z x}$
Multiplying all three inequalities, we get
$(x + y) (y + z) (z + x) \geq 8 x y z$
Since $x + y + z = 1,$
$∴ (1 - z) (1 - x) (1 - y) \geq 8 x y z$
$\Rightarrow \frac{(1 - z) (1 - x) (1 - y)}{x y z} \geq 8$
$∴$ Least value $= 8$ and it occurs when $x = y = z = \frac{1}{3}$

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If x,y,z∈(0,1) such that x+y+z=1, then the least value of (1−x)(1−y)(1−z)xyz is

The correct option is A 8Using A.M. ≥ G.M., we get x+y≥2√xy, y+z≥2√yz, z+x≥2√zx Multiplying all three inequalities, we get (x+y)(y+z)(z+x)≥8xyz Since x+y+z=1, ∴(1−z)(1−x)(1−y)≥8xyz ⇒(1−z)(1−x)(1−y)xyz≥8 ∴ Least value =8 and it occurs when x=y=z=13

If $x, y, z \in (0, 1)$ such that $x + y + z = 1,$ then the least value of $\frac{(1 - x) (1 - y) (1 - z)}{x y z}$ is