Let $g(x) = ||x + 2| - 3|$. If $a$ denotes the number of relative minima,$b$ denotes the number of relative maxima,and $c$ denotes the product of the zeroes of $g(x)$,then the value of $(a + 2b - c)$ is:

Let $S$ be a finite set. Then a non-identity function $f: S \rightarrow S$ can be

Consider the function $f: [\frac{1}{2}, 1] \rightarrow \mathbb{R}$ defined by $f(x) = 4\sqrt{2}x^3 - 3\sqrt{2}x - 1$. Consider the following statements:
$(I)$ The curve $y = f(x)$ intersects the $x$-axis exactly at one point.
$(II)$ The curve $y = f(x)$ intersects the $x$-axis at $x = \cos \frac{\pi}{12}$.
Then:

If $f(x)=\frac{2^{x}+2^{-x}}{2}$,then $f(x+y) \cdot f(x-y)$ is

The function $f(x) = \sec \left[ \log \left( x + \sqrt{1 + x^2} \right) \right]$ is . . . . . . function.

Let $A = \{1, 2, 3, 5, 8, 9\}$. Then the number of possible functions $f : A \rightarrow A$ such that $f(m \cdot n) = f(m) \cdot f(n)$ for every $m, n \in A$ with $m \cdot n \in A$ is equal to $...............$.