If $f(x) = \sin \log x$, then the value of $f(xy) + f\left( {\frac{x}{y}} \right) – 2f(x).\cos \log y$ is equal to

If $f(x)=\frac{\left(\tan 1^{\circ}\right) x+\log _{\varepsilon}(123)}{x \log _{\varepsilon}(1234)-\left(\tan 1^{\circ}\right)}, x > 0$, then the least value of $f(f(x))+f\left(f\left(\frac{4}{x}\right)\right)$ is $………..$.

Solve $|x\,-\,2| + |x\,-\,1| = x\,-\,3$

Which of the following is correct

Let $N$ be the set of positive integers. For all $n \in N$, let $f_n=(n+1)^{1 / 3}-n^{1 / 3} \text { and }$ $A=\left\{n \in N: f_{n+1}<\frac{1}{3(n+1)^{2 / 3}} < f_n\right\}$ Then,