If $f(x)$ is a function satisfying the condition $f(x) = \frac{1}{3}\left[ f(x + 6) + \frac{6}{f(x + 7)} \right]$ and $f(x) \geq 0$ for all $x \in R$. If $\lim_{x \to \infty} f(x) = \sqrt{m}$,then the value of $m$ is:

Let $f$ be a function satisfying $f(xy) = \frac{f(x)}{y}$ for all positive real numbers $x$ and $y.$ If $f(30) = 20,$ then the value of $f(40)$ is-

Let $a, b, c \in \mathbb{R}$. If $f(x) = ax^2 + bx + c$ is such that $a + b + c = 3$ and $f(x + y) = f(x) + f(y) + xy, \forall x, y \in \mathbb{R}$,then $\sum_{n=1}^{10} f(n)$ is equal to:

Let $f(x + y) = f(x) + f(y)$ for all $x, y \in R.$ Then:

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: