Using $A.M. \ge G.M.$,
$\frac{{{e^x} + {e^{ - x}}}}{2} \ge \sqrt {{e^x}.{e^{ - x}}} $
$ \Rightarrow {e^x} + {e^{ - x}} \ge 2$
$\therefore \sin ({\pi ^x}) \ge 2$ which is not possible for any real x.
So, no solution.
If the sum of the coefficients in the expansion of $(x+y)^n$ is 4096, then the greatest coefficient in the expansion is _ _ _ _ . Solution $C_0 + C_1 + C_2 + C_3 + ......................... + C_n =4096 $ $\therefore 2^n = 4096 =2^{12} $ $\Rightarrow n = 12 $ Greatest coefficient = ${}^{12}{C_6} = 924$