There are $50$ apple trees in an orchard. Each tree produces $800$ apples. For each additional tree planted in the orchard,the output per tree drops by $10$ apples. The number of trees that should be added to the existing orchard to maximize the total output is:

If the function $f(x) = \begin{cases} \tan^{-1}x; & x < 1 \\ \sec^{-1}x + \lambda; & x \ge 1 \end{cases}$ has a local minimum at $x = 1$,then the range of $\lambda$ is:

$A$ wire of length $36 \text{ m}$ is cut into two pieces. One piece is bent to form a square and the other is bent to form a circle. If the sum of the areas of the two figures is minimum,and the circumference of the circle is $k \text{ m}$,then $\left(\frac{4}{\pi}+1\right) k$ is equal to ..... .

The maximum value of ${\left( {\frac{1}{x}} \right)^x}$ is

The number of critical points of the function $f(x)=(x-2)^{2/3}(2x+1)$ is:

The two positive numbers with sum $t$,and the sum of their squares is minimum are