Quadratic polynomial

Consider a quadratic function $f(x)=a x^2+b x+c$ where $a,b,c \in R$ are real no.'s and $a$ is a non zero number and satisfy the following condition.

$(i)$ $f(x-4)=f(x-2)$ and $f(x)\geq x$ where $x\in R$ .

$(ii)$ $f(x)\leq (\frac{x+1}{2})^2$ where $x$ $\in$ $(0,2)$

$(iii)$ The minimum value of $f(x)$ is zero

$Q-1$ . The value of leading coefficient of quadratic polynomial is?

$Q-2$ . $f'(1)$ has the value equal to?

If the answers of $Q-1$ and $Q-2$ are $'a'$ and $'b'$ respectively then the value of $a\times b$ is

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