一道高数题设函数f(x)在[o,1]上具有二阶导数,具满足条件|f(x)|<=a,|f"(x)|<=b.

其中a,b都是非负常数,c是(0,1)内任意一点,证明|f✀(c)|&lt;=2a (b/2)
2025-05-14 23:35:32
推荐回答(1个)
回答1:

f(0)=f(c)-f'(c)*c+f''(m)*c^2/2
f(1)=f(c)+f'(c)*(1-c)+f''(n)*(1-c)^2/2
两式相减,得
f'(c)=f(1)-f(0)-f''(m)*c^2/2+f''(n)*(1-c)^2/2
所以
|f'(c)|<|f(1)|+|f(0)|+|f''(m)|*c^2/2+|f''(n)|*(1-c)^2/2
<2a+(b/2)