Let f(x) be a second degree polynomial. If f( | vedclass

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(A) Let the second degree polynomial be $f(x) = ax^2 + bx + c$,where $a 
eq 0$. Given $f(1) = f(-1)$,we have $a(1)^2 + b(1) + c = a(-1)^2 + b(-1) + c

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in $A$.$P$.

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(A) Let the second degree polynomial be $f(x) = ax^2 + bx + c$,where $a 
eq 0$. Given $f(1) = f(-1)$,we have $a(1)^2 + b(1) + c = a(-1)^2 + b(-1) + c$,which simplifies to $a + b + c = a - b + c$,implying $2b = 0$,so $b = 0$. Thus,$f(x) = ax^2 + c$. The derivative is $f^{\prime}(x) = 2ax$. Since $p, q, r$ are in $A$.$P$.,we have $2q = p + r$. Multiplying by $2a$,we get $2a(2q) = 2a(p + r)$,which means $2(2aq) = 2ap + 2ar$. Substituting $f^{\prime}(x) = 2ax$,we get $2f^{\prime}(q) = f^{\prime}(p) + f^{\prime}(r)$. This condition implies that $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are in $A$.$P$.

Let $f(x)$ be a second degree polynomial. If $f(1) = f(-1)$ and $p, q, r$ are in $A$.$P$.,then $f^{\prime}(p), f^{\prime}(q), f^{\prime}(r)$ are

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