The maximum value of $f(x) = \frac{\log x}{x}$ $(x > 0, x \neq 1)$ is

If for a function $f(x)$,$f'(a) = 0$ and $f''(a) = 0$,then $x = a$ is:

The number of real roots of the equation $e^{4x} + 2e^{3x} - e^{x} - 6 = 0$ is:

$A$ wire of length $8 \text{ units}$ is cut into two parts which are bent respectively in the form of a square and a circle. The least value of the sum of the areas so formed is

Let $A$ be the region enclosed by the parabola $y^2=2x$ and the line $x=24$. Then the maximum area of the rectangle inscribed in the region $A$ is ..................

Let $(2\alpha, \alpha)$ be the largest interval in which the function $f(t) = \frac{|t+1|}{t^2}, t < 0$, is strictly decreasing. Then the local maximum value of the function $g(x) = 2\log_e(x-2) + \alpha x^2 + 4x - \alpha, x > 2$, is