the minimum value of x 4 y 4 z 2xyz for positive real

The minimum value of $$\frac{{{x^4} + {y^4} + {z^2}}}{{xyz}}$$ for positive real number $$x, y, z$$ is

A. $$\sqrt 2 $$

B. $$2\sqrt 2 $$

C. $$4\sqrt 2 $$

D. $$8\sqrt 2 $$

Answer : $$2\sqrt 2 $$

Solution :
By $${\text{A}}{\text{.M}}{\text{.}} \geqslant {\text{G}}{\text{.M}}{\text{.}}$$
$$\eqalign{ & {x^4} + {y^4} \geqslant 2{x^2}{y^2}{\text{ and }}2{x^2}{y^2} + {z^2} \geqslant \sqrt 8 \,xyz. \cr & \Rightarrow \frac{{{x^4} + {y^4} + {z^2}}}{{xyz}} \geqslant \sqrt 8 \cr} $$

Releted MCQ Question on
Algebra >> Sequences and Series

Releted Question 1

If $$x, y$$ and $$z$$ are $${p^{{\text{th}}}},{q^{{\text{th}}}}\,{\text{and }}{r^{{\text{th}}}}$$ terms respectively of an A.P. and also of a G.P., then $${x^{y - z}}{y^{z - x}}{z^{x - y}}$$ is equal to:

A. $$xyz$$

B. 0

C. 1

D. None of these

View Answer

Releted Question 2

The third term of a geometric progression is 4. The product of the first five terms is

A. $${4^3}$$

B. $${4^5}$$

C. $${4^4}$$

D. none of these

View Answer

Releted Question 3

The rational number, which equals the number $$2.\overline {357} $$ with recurring decimal is

A. $$\frac{{2355}}{{1001}}$$

B. $$\frac{{2379}}{{997}}$$

C. $$\frac{{2355}}{{999}}$$

D. none of these

View Answer

Releted Question 4

If $$a, b, c$$ are in G.P., then the equations $$a{x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $$\frac{d}{a},\frac{e}{b},\frac{f}{c}$$ are in-

A. A.P.

B. G.P.

C. H.P.

D. none of these

View Answer

The minimum value of $$\frac{{{x^4} + {y^4} + {z^2}}}{{xyz}}$$ for positive real number $$x, y, z$$ is

Releted MCQ Question on
Algebra >> Sequences and Series

If $$x, y$$ and $$z$$ are $${p^{{\text{th}}}},{q^{{\text{th}}}}\,{\text{and }}{r^{{\text{th}}}}$$ terms respectively of an A.P. and also of a G.P., then $${x^{y - z}}{y^{z - x}}{z^{x - y}}$$ is equal to:

The third term of a geometric progression is 4. The product of the first five terms is

The rational number, which equals the number $$2.\overline {357} $$ with recurring decimal is

If $$a, b, c$$ are in G.P., then the equations $$a{x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $$\frac{d}{a},\frac{e}{b},\frac{f}{c}$$ are in-

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Sequences and Series

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The minimum value of $$\frac{{{x^4} + {y^4} + {z^2}}}{{xyz}}$$ for positive real number $$x, y, z$$ is

Releted MCQ Question on Algebra >> Sequences and Series

If $$x, y$$ and $$z$$ are $${p^{{\text{th}}}},{q^{{\text{th}}}}\,{\text{and }}{r^{{\text{th}}}}$$ terms respectively of an A.P. and also of a G.P., then $${x^{y - z}}{y^{z - x}}{z^{x - y}}$$ is equal to:

The third term of a geometric progression is 4. The product of the first five terms is

The rational number, which equals the number $$2.\overline {357} $$ with recurring decimal is

If $$a, b, c$$ are in G.P., then the equations $$a{x^2} + 2bx + c = 0$$ and $$d{x^2} + 2ex + f = 0$$ have a common root if $$\frac{d}{a},\frac{e}{b},\frac{f}{c}$$ are in-

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Releted MCQ Question on
Algebra >> Sequences and Series

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Sequences and Series