f'(x)=1-1/(x-1)².
設(shè)x=t點(diǎn)處的切線為y=f'(t)(x-t)+f(t)
即y=[1-1/(t-1)²](x-t)+t+1/(t-1)=x-x/(t-1)²+t/(t-1)²+1/(t-1)
它與x=1的交點(diǎn)：將x=1代入有
y=1-1/(t-1)²+t/(t-1)²+1/(t-1)=1+(-1+t+t-1)/(t-1)²=1+(2t-2)/(t-1)²=1+2/(t-1),
即交點(diǎn)在(1,1+2/(t-1)),
它與y=x的交點(diǎn)：將y=x代入有x=x-x/(t-1)²+t/(t-1)²+1/(t-1)有x=(t-1)²[t/(t-1)²+1/(t-1)]=t+t-1=2t-1,
交點(diǎn)在(2t-1,2t-1),
y=x與x=1的交點(diǎn)在(1,1),
所以該三角形的面積S=1/2×|[1+2/(t-1)-1]×[2t-1-1]|=1/2×|2/(t-1) ×2(t-1)|=2為定值.