Question

已知f(x)是一元二次函数,若f(0)=0.且有f(x+1)=f(x)+x+1,求f(x)的解析式

Answer 1

因为f(x)是一元二次函数

所以设 f(x) = ax² + bx + c ( a ≠ 0)

因为f(x + 1) = f( x ) + x + 1

a(x + 1)² + b(x + 1) + c = ax² + bx + c + x + 1

2ax + a + b = x + 1

所以 2a = 1 ,a + b = 1

所以 a = 1/2 ,b = 1/2

又因为f(0) = 0

所以 c = 0

所以 f(x) = x²/2 + x/2