Question

Biết y = f(x) = a\(x^2\) + bx +c là hàm số thoả mãn: f(x) + 2f(1-x) = 3\(x^2\) + 2x +5, mọi x thuộc R. Tính f(6)

Answer 1

\(f\left(1-x\right)=a\left(1-x\right)^2+b\left(1-x\right)+c=ax^2-\left(2a+b\right)x+a+b+c\)

\(\Rightarrow f\left(x\right)+2f\left(1-x\right)=3ax^2-\left(4a+b\right)x+2a+2b+3c\)

\(f\left(x\right)+2f\left(1-x\right)=3x^2+2x+5\) \(\forall x\)

\(\Leftrightarrow\left\{{}\begin{matrix}3a=3\\4a+b=-2\\2a+2b+3c=5\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}a=1\\b=-6\\c=5\end{matrix}\right.\)

\(\Rightarrow f\left(x\right)=x^2-6x+5\Rightarrow f\left(6\right)=6^2-6.6+5=5\)