设函数f(x)为二次函数,且f(x+1）+f(x-1)=x∧2-2x+6,求f(x)

题目

设函数f(x)为二次函数,且f(x+1）+f(x-1)=x∧2-2x+6,求f(x)

答案

f(x)=ax²+bx+cf(x+1)+f(x-1)=a(x+1)²+b(x+1)+c+a(x-1)²+b(x-1)+c=2ax²+2bx+(2a+2c)=x²-2x+62a=12b=-22a+2c=6a=1/2,b=-1,c=5/2f(x)=x²/2-x+5/2