The number of distinct real roots of the equation $3x^{4} + 4x^{3} - 12x^{2} + 4 = 0$ is ..... .

$x$ 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 $\frac{\alpha}{2} + \frac{\beta}{5} =$

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 $f(x) = x + \frac{1}{x}$,$x \neq 0$,then the local maximum and minimum values of the function $f$ are respectively....

If $y = a \ln |x + 1| + b(x + 1)^2 + x$ has an extremum value of $4$ at $x = 0$,then the value of $(a, b)$ is:

The function,$f(x)=x \sqrt{1-x}$,where $x \in(0,1)$,has a local maximum at $x=$