If $A + B = \frac{\pi}{2}$,what is the maximum value of $\cos A \cos B$?

Given $a^2 + 2a + \csc^2 \left( \frac{\pi}{2}(a + x) \right) = 0$,then which of the following holds good?

The minimum value of $f(x) = a^{a^{x}} + a^{1-a^{x}}$,where $a, x \in R$ and $a > 0$,is equal to ..... .

The maximum value of the function $f(x) = 3 \sin x + 4 \cos x$ is

The maximum value of $(\cos \alpha_1) \cdot (\cos \alpha_2) \ldots (\cos \alpha_n)$ under the constraints $0 \leq \alpha_1, \alpha_2, \ldots, \alpha_n \leq \frac{\pi}{2}$ and $(\cot \alpha_1) \cdot (\cot \alpha_2) \ldots (\cot \alpha_n) = 1$ is

What is the minimum value of $a \sec x + b \csc x$ for $0 < a < b$ and $0 < x < \pi/2$?