Question

对函数f(x)=sinx及F(x)=x+cosx在区间[0,π/2]上验证柯西中值定理的正确性

Answer 1

f(0)=0, f(π/2)=1

F(0)=1, F(π/2)=π/2

f'(x)=cosx, F'(x)=1-sinx

[f(0)-f(π/2)]/[F(0)-F(π/2)]=1/(π/2-1)

令f'(t)/F'(t)=cost/(1-sint)=1/(π/2-1)

即2-2sint=(π-2)cost,此方程在[0,π/2]有唯一根.

解得：sint(t+p)=1/√[1+(π-2)^2/4]=0.868479575, 其中tanp=(π-2)/2, p=0.518669369

因此有：t=0.533457588 ∈[0,π/2] 满足柯西中值定理.