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

Let $f(x) = \alpha x^2 - 2 + \frac{1}{x}$,where $\alpha$ is a real constant. The smallest $\alpha$ for which $f(x) \geq 0$ for all $x > 0$ is

For $a > 0$,if the function $f(x) = 2x^3 - 9ax^2 + 12a^2x + 1$ attains its maximum value at $p$ and minimum value at $q$ such that $p^2 = q$,then $a =$

Let $f(x) = \begin{cases} 3 - x, & 0 \le x < 1 \\ x^2 + \log_e b, & x \ge 1 \end{cases}$. The set of values of $b$ such that $f(x)$ has a local minimum at $x = 1$ is

For which values of $x$ is the function $f(x) = \sin x + \cos 2x$ $(x > 0)$ minimum?

The minimum value of $f(x) = x + \frac{4}{x + 2}$ for $x > -2$ is