Algebra Level pending

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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