f(x)二阶连续可导,f''(x)不等于0f(x+kh)=f(x)+f'(x+kh)hk大于0小于1,证明limh趋于0k=1/2

Question

f(x)二阶连续可导,f ''(x)不等于0 f(x+kh)=f(x)+f '(x+kh)h k大于0小于1,证明lim h趋于0 k=1/2

Answer 1

你写得题目不对.等号前面是f(x+h),不是f(x+kh).
做Taylor展式：f(x+h)=f(x)+f'(x)h+0.5f''(x)h^2+小o(h^2),（*）
f'(x+kh)=f‘(x)+f''(x)kh+小o(h),将此代入原题表达式得
f(x+h)=f(x)+f'(x)h+f''(x)kh^2+小o(h^2),与(*)式比较得0.5f''(x)h^2+小o(h^2)=f''(x)kh^2+小o(h^2),两边同除以h^2,在令h趋于0,得0.5f''(x)=f''(x)*lim k,因此lim k=1/2