Let $f: R \rightarrow R$ be such that for all $\mathrm{x} \in \mathrm{R}\left(2^{1+\mathrm{x}}+2^{1-\mathrm{x}}\right), f(\mathrm{x})$ and $\left(3 ^\mathrm{x}+3^{-\mathrm{x}}\right)$ are in $A.P.$, then the minimum value of $f(x)$ is

If the product of three terms of $G.P.$ is $512$. If $8$ added to first and $6$ added to second term, so that number may be in $A.P.$, then the numbers are

If $a,\,b,\,c,\,d$ are positive real numbers such that $a + b + c + d$ $ = 2,$ then $M = (a + b)(c + d)$ satisfies the relation

Let $x, y, z$  be positive real numbers such that $x + y + z = 12$ and  $x^3y^4z^5 = (0. 1 ) (600)^3$. Then $x^3 + y^3 + z^3$ is equal to

Let $b_i>1$ for $i=1,2, \ldots, 101$. Suppose $\log _e b_1, \log _e b_2, \ldots, \log _e b_{101}$ are in Arithmetic Progression ($A.P$.) with the common difference $\log _e 2$. Suppose $a_1, a_2, \ldots, a_{101}$ are in $A.P$. such that $a_1=b_1$ and $a_{51}=b_{51}$. If $t=b_1+b_2+\cdots+b_{51}$ and $s=a_1+a_2+\cdots+t_{65}$, then

The $A.M., H.M.$ and $G.M.$ between two numbers are $\frac{{144}}{{15}}$, $15$ and $12$, but not necessarily in this order. Then $H.M., G.M.$ and $A.M.$ respectively are