If $f(x + ay, x - ay) = axy$,then $f(x, y)$ is equal to

Let $f: R \rightarrow R$ be defined by $f\left(\frac{x+y}{2}\right)=\frac{f(x)+f(y)}{2}$ for all $x$ and $y$. If $f^{\prime}(0)$ exists and equals $-1$ and $f(0)=1$, then $f(2)=$

Let $R$ denote the set of all real numbers. Let $a_i, b_i \in R$ for $i \in \{1, 2, 3\}$. Define the functions $f: R \rightarrow R$,$g: R \rightarrow R$,and $h: R \rightarrow R$ by $f(x) = a_1 + 10x + a_2x^2 + a_3x^3 + x^4$ and $g(x) = b_1 + 3x + b_2x^2 + b_3x^3 + x^4$. Let $h(x) = f(x+1) - g(x+2)$. If $f(x) \neq g(x)$ for every $x \in R$,then the coefficient of $x^3$ in $h(x)$ is:

$f: R \rightarrow R$ is a function such that $f(0)=1$ and for all $x, y \in R$,$f(xy+1)=f(x)f(y)-f(y)-x+2$. Then $\frac{df}{dx}$ at $x=e$ is:

Let $f: R \rightarrow R$ be a differentiable function that satisfies the relation $f(x + y) = f(x) + f(y) - 1$ for all $x, y \in R$. If $f'(0) = 2$,then $|f(-2)|$ is equal to:

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: