If $f(x)$ is a polynomial function satisfying $f(x) \cdot f(\frac{1}{x}) = f(x) + f(\frac{1}{x})$ and $f(4) = 65$,then the value of $f(6)$ is:

Let $f: R \rightarrow R$ be a twice differentiable function such that $(\sin x \cos y)(f(2x+2y) - f(2x-2y)) = (\cos x \sin y)(f(2x+2y) + f(2x-2y))$ for all $x, y \in R$. If $f'(0) = \frac{1}{2}$,then the value of $24f''\left(\frac{5\pi}{3}\right)$ is:

Let $f: N \rightarrow N$ be a function such that $f(m+n)=f(m)+f(n)$ for every $m, n \in N$. If $f(6)=18$ then $f(2) \cdot f(3)$ is equal to :

If $f : R \to R$ is such that $f(x + y) = f(x) + f(y)$ for all $x, y \in R$,$f(1) = 7$ and $\sum_{r=1}^{n} f(r) = 14112$,then $n$ is equal to:

Let $f: R \rightarrow R$ be a function which satisfies $f(x+y)=f(x)+f(y)$ for all $x, y \in R$. If $f(1)=2$ and $g(n)=\sum_{k=1}^{n-1} f(k)$ for $n \in N$,then the value of $n$ for which $g(n)=20$ is:

If $(f(x))^2 = f(x^2) + f(1)$ holds good,then find $f(x)$.