A.M., G.M., H.M. and Their Relations

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

Let $A\mathit{,}\mathit{ }G$ and $H$ be the arithmetic mean, geometric mean and harmonic mean, respectively of two distinct positive real numbers. If $\alpha$ is the smallest of the two roots of the equation $A\left(G-H\right)x^2+G\left(H-A\right)x+H\left(A-G\right)=0,$ then 

Let $x$ and $y$ be positive real numbers. What is the smallest possible value of $\frac{16}{x}+\frac{108}{y}+xy$?

The function $f\left(x\right)=\sin x+\frac{1}{\sin x}$ in the interval $\left(0, \pi \right)$ has a

The minimum value of $9{\tan }^2\theta +4{\cot }^2\theta$ is _________.

Two numbers $x$ and $y$ have arithmetic mean $9$ and geometric mean $4.$ Then, $x$ and $y$ are the roots of

The minimum value of the quantity $\frac{\left({\mathrm{a}}^2+3\mathrm{a}+1\right)\mathrm{ }\left({\mathrm{b}}^2+3\mathrm{b}+1\right)\mathrm{ }\left({\mathrm{c}}^2+3\mathrm{c}+1\right)}{\mathrm{a}\mathrm{b}\mathrm{c}}$ where $\mathrm{a},\mathrm{ }\mathrm{b},\mathrm{ }\mathrm{c}\in \mathrm{ }{\mathrm{R}}^{+}$ is-

If $f\left(\theta \right)=2({\sec }^2\theta +{\cos }^2\theta )$$,$ then its value always

If the arithmetic mean of two numbers $a$ and $b, a>b>0$ , is five times their geometric mean, then $\frac{a+b}{a-b}$ is equal to:

If three positive numbers $a$, $b$ and $c$ are in A.P. such that $abc=8$, then the minimum possible value of $b$ is:

Find the minimum value of $\frac{4+9x^2{\sin }^{2}x}{x\sin x}$ for $0<x<\pi$.

If ${\log }_{6}x+{\log }_{6}y\ge 4$ then the smallest possible value of $x+y$ 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