Relationship Between Arithmetic Mean and Geometric Mean

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

Find the greatest value of $x^3{y}^4, \text{if} 2x+3y=7 \text{and} x⩾0, y⩾0.$

${16}^{x^2+y}+{16}^{x+y^2}=1$. Solve for all $x,y\in \mathrm{ℝ}$

If $a,b,c$ are positive real numbers such that $\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=1$, then maximum value of $64 abc$ is 

The minimum value of $\frac{\left(x^2+x+1\right)\left(y^2+y+1\right)}{xy}$ where $x,y >0$ is

If $a,b,c$ are non-zero real numbers, then the minimum value of the expression $\frac{\left(a^8+4a^4+1\right)\left(b^8+3b^4+1\right)\left(c^2+2c+2\right)}{a^4{b}^4}$ equals

If A.M. and G.M. of two positive numbers $a$ and $b$ are $10$ and $8$, respectively, find the numbers.

Let $a,b,c>0, a+b+c=15$. The least value of $E=\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\text{ is}$________.

The sum of square of any real positive quantities and its reciprocal is never less

If $f\left(x\right)=\frac{3x^2+1}{\sqrt{x^4+x^2}}\forall 0<x\le \sqrt{2}$, then $f\left(x\right)$ has minimum value $M$ at $x=m$. Then, $\left(m^2+\frac{1}{M^2}\right)$ is equal to

If A.M of two numbers is $11$ and their G.M. is $2\sqrt{30}$, find the two numbers.
 

If A.M of two numbers be twice their G.M then the numbers are in the ratio.

If $a,b,c,d$ denote the sides of a quadrilateral $ABCD$ then, $\frac{a}{b+c+d}+\frac{b}{c+d+a}+\frac{c}{a+b+d}+\frac{d}{a+b+c}$ is

The minimum value of ${\tan }^2\mathrm{x}+{\cot }^2\mathrm{x}$ is 

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 _________.

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 $a, b, c, d$ are four positive real numbers such that $abcd = 1,$ then the minimum value of the expression $\left(1+a\right)\left(1+b\right) \left(1+c\right) \left(1+d\right)$ is equal to

Find the maximum value for the function $y=\frac{x}{a{x}^2+b}(a,b>0)$.