The local maximum value of $f(x) = x + \frac{1}{x}$ for $x < 0$ is equal to:

If $xy = c^2$,then the minimum value of $ax + by$ is

If $\overline{PQ}$ and $\overline{PR}$ are two sides of $\Delta PQR$,for what value of the angle between $\overline{PQ}$ and $\overline{PR}$ is the area of the triangle maximum?

$A$ right triangle is drawn in a semicircle of radius $R = \frac{1}{2}$ with one of its legs along the diameter. The maximum area of the triangle is

If the function $f(x) = a \sin(x) + \frac{1}{3} \sin(3x)$ attains its maximum value at $x = \frac{\pi}{3}$,then $a$ equals:

If the function $f(x) = x^3 - 3(a - 2)x^2 + 3ax + 7$,for some $a \in R$,is increasing in $(0, 1]$ and decreasing in $[1, 5)$,then a root of the equation $\frac{f(x) - 14}{(x - 1)^2} = 0$ $(x \neq 1)$ is