Relation between A.M., G.M., H.M.

If $a_1,a_2,.....a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+a_3+.......+a_{n-1}+2a_n$ is

If $a_1, a_2, .......a_n$ are positive real numbers whose product is a fixed number $e$, the minimum value of $a_1+a_2+ .......+2a_n$ is

If $xyz=2$ and $x, y, z>0$, then minimum value of $\left(2+x\right) \left(2+y\right) \left(2+z\right)$ is

If $f\left(x\right)={\left(\sin x+\mathrm{cosec}x\right)}^2+{\left(\cos x+\sec x\right)}^2+{\left(\tan x+\cot x\right)}^2-3$, then the minimum value of $f\left(x\right)$ is 

If $a,b,c$ are positive real numbers, then minimum value of $\frac{\left(a+b+c\right)\left(a^2+b^2+c^2\right)}{abc}$ is

Let $x$ and $y$ be two positive real numbers such that $x+y=1$. Then the minimum value of $\frac{1}{x}+\frac{1}{y}$ is-

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

The number of three digit numbers $\overline{abc}$ such that the arithmetic mean of $b \& c$ and the square of their geometric mean are equal is

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

 The greatest value of $\frac{\text{xyz}}{\left(\text{x}-\text{m}\right)\left(\text{y}-\text{n}\right)\left(\text{z}-\text{r}\right)}$  is 

If $x>m\; ,\; y>n\; ,\; z>\text{r}(x,\; y,\; z\; >\; 0)$ such that $\left|\begin{array}{ccc}\text{x}& \text{n}& \text{r}\\ \text{m}& \text{y}& \text{r}\\ \text{m}& \text{n}& \text{z}\end{array}\right|=0$.

The value of $\frac{\text{m}}{\text{x}-\text{m}}+\frac{\text{y}}{\text{y}-\text{n}}+\frac{\text{r}}{\text{z}-\text{r}}$  is

If $x>\mathrm{m},\mathrm{y}>\mathrm{n},\mathrm{z}>r(\mathrm{x},\mathrm{y,}\mathrm{z\; >\; 0)}$ such that $\left|\begin{array}{ccc}x& n& r\\ m& y& r\\ m& n& z\end{array}\right|=0$. The value of $\frac{x}{x-m}+\frac{y}{y-n}+\frac{z}{z-r}$, is

If $x, y\in (0,\infty )$, then the least possible value of $x^{100}+\frac{100}{x}+y^{50}+\frac{50}{y}$ is

If $x$ and $y$ are positive real numbers and $m, n$ are any positive integers then $\frac{x^n{y}^m}{\left(1+x^{2n}\right)\left(1+y^{2m}\right)}$ can be :

If the first and ${\left(2n-1\right)}^{th}$terms of an $A.P, G.P \text{and} H.P$ with positive terms are equal and their $n^{th}$ terms are $a, b, c$ respectively, then which of the following options must be correct: 

Let $a, x, b$ be in $AP ; a, y, b$ are in $GP$ and $a, z, b$ are in $HP$ where $a$ and $b$ are distinct positive real number. If $x=y+2$ and $a=5z$, then :

If $a_i>0$ for all $i\in N$, then :

Let $1, abc,$ $a^2{b}^2{c}^2$ are in $A.P.$ $(a, b, c>0),$ then minimum value of $27 a+8 b+125 c$ is :

Statement $1$: $27abc\ge {\left(a+b+c\right)}^3$ and $3a+4b+5c=12$ then $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}=10,$ where$a, b, c$  positive real number.
Statement $2$: For positive numbers, $\mathrm{AM}>\mathrm{GM}$.

If $a, b \& c$ are positive real numbers and $\{\left(1+a\right)\left(1+b\right)\left(1+c\right){\}}^7>k^7{a}^4{b}^4{c}^4$, then maximum integer value of $k$ is

If $\lambda , \mu$ be real numbers such that $x^3-\lambda x^2+\mu x-6=0$ has its roots real and positive then the minimum value of $\mu$ is $\frac{6^{{\displaystyle \frac{5}{3}}}}{\lambda }$, then the value of $\lambda$ is