$x$ and $y$ are two positive integers such that $2x + 3y = 50$. If $x^2 y^3$ is maximum for $x = \alpha$ and $y = \beta$, then $\frac{\alpha}{2} + \frac{\beta}{5} =$

The stationary points of the curve $y = 8x^2 - x^4 - 4$ are . . . . . .

Let $f(x) = \frac{\sin \pi x}{x^2}, x > 0$. Let $x_1 < x_2 < x_3 < \ldots < x_n < \ldots$ be all the points of local maximum of $f(x)$ and $y_1 < y_2 < y_3 < \ldots < y_n < \ldots$ be all the points of local minimum of $f(x)$. Which of the following statements are true?
$(1)$ $|x_n - y_n| > 1$ for every $n$
$(2)$ $x_1 < y_1$
$(3)$ $x_n \in (2n, 2n + \frac{1}{2})$ for every $n$
$(4)$ $x_{n+1} - x_n > 2$ for every $n$

The minimum value of $\frac{(5 + x)(2 + x)}{1 + x}$ for non-negative real $x$ is

Find the maximum and minimum values of the function given by $g(x) = x^{3} + 1$.

If $f(x) = x + \frac{1}{x}$ for $x > 0$,then its greatest value is: