Question

确定常数a,b,使得∫[0,1] [f(x)-(a+bx)]^2 dx最小?

Answer 1

I(a,b)=∫[0,1] [f(x)-(a+bx)]^2 dx;

Ia(a,b)=-2∫[0,1] [f(x)-(a+bx)] dx=0,

Ib(a,b)=-2∫[0,1] [f(x)-(a+bx)]x dx=0;

2a+b=2∫[0,1] f(x)dx,

3a+2b=6∫[0,1] xf(x)dx;

a=4∫[0,1] f(x)dx-6∫[0,1]x f(x)dx,

b=-6∫[0,1] f(x)dx+12∫[0,1] xf(x)dx.