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

$[t]$ denotes the greatest integer function and $[t-m] = [t] - m$ when $m \in \mathbb{Z}$. If $k = 2[2x - 1] - 1$ and $3[2x - 2] + 1 = 2[2x - 1] - 1$,then the range of $f(x) = [k + 5x]$ is

If $a+\alpha=1, b+\beta=2$ and $af(x)+\alpha f\left(\frac{1}{x}\right)=bx+\frac{\beta}{x}$ for $x \neq 0$,then the value of the expression $\frac{f(x)+f\left(\frac{1}{x}\right)}{x+\frac{1}{x}}$ is ..... .

There are three kinds of liquids $X, Y, Z$. Three jars $J_1, J_2, J_3$ contain $100 \, ml$ of liquids $X, Y, Z$ respectively. An operation consists of three steps in the following order:
- Stir the liquid in $J_1$ and transfer $10 \, ml$ from $J_1$ into $J_2$.
- Stir the liquid in $J_2$ and transfer $10 \, ml$ from $J_2$ into $J_3$.
- Stir the liquid in $J_3$ and transfer $10 \, ml$ from $J_3$ into $J_1$.
After performing the operation four times,let $x, y, z$ be the amounts of $X, Y, Z$ respectively in $J_1$. Then,

Let $f:[0,1] \rightarrow \mathbb{R}$ be an injective continuous function that satisfies the condition $-1 < f(0) < f(1) < 1$. Then,the number of functions $g:[-1,1] \rightarrow [0,1]$ such that $(g \circ f)(x) = x$ for all $x \in [0,1]$ is

Given $f'(x) > 0$ and $g'(x) < 0$ for all $x \in R$,then which of the following is true?