设函数f(x)在[0,1]上具有三节连续导数且f(0)=1, f(1)=2, f'(1/2)=0.证明:(0,1)内至少存在一点a,使│f'''(a)│≥24.请问这题怎么做?谢谢了……
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![设函数f(x)在[0,1]上具有三节连续导数且f(0)=1, f(1)=2, f'(1/2)=0.证明:(0,1)内至少存在一点a,使│f'''(a)│≥24.请问这题怎么做?谢谢了……](/uploads/image/z/1042294-22-4.jpg?t=%E8%AE%BE%E5%87%BD%E6%95%B0f%28x%29%E5%9C%A8%5B0%2C1%5D%E4%B8%8A%E5%85%B7%E6%9C%89%E4%B8%89%E8%8A%82%E8%BF%9E%E7%BB%AD%E5%AF%BC%E6%95%B0%E4%B8%94f%280%29%3D1%2C+f%281%29%3D2%2C+f%27%281%2F2%29%3D0.%E8%AF%81%E6%98%8E%EF%BC%9A%EF%BC%880%2C1%EF%BC%89%E5%86%85%E8%87%B3%E5%B0%91%E5%AD%98%E5%9C%A8%E4%B8%80%E7%82%B9a%2C%E4%BD%BF%E2%94%82f%27%27%27%28a%29%E2%94%82%E2%89%A524.%E8%AF%B7%E9%97%AE%E8%BF%99%E9%A2%98%E6%80%8E%E4%B9%88%E5%81%9A%3F%E8%B0%A2%E8%B0%A2%E4%BA%86%E2%80%A6%E2%80%A6)
设函数f(x)在[0,1]上具有三节连续导数且f(0)=1, f(1)=2, f'(1/2)=0.证明:(0,1)内至少存在一点a,使│f'''(a)│≥24.请问这题怎么做?谢谢了……
设函数f(x)在[0,1]上具有三节连续导数且f(0)=1, f(1)=2, f'(1/2)=0.证明:(0,1)内至少存在一点a,使│f'''(a)│≥24.
请问这题怎么做?谢谢了……
设函数f(x)在[0,1]上具有三节连续导数且f(0)=1, f(1)=2, f'(1/2)=0.证明:(0,1)内至少存在一点a,使│f'''(a)│≥24.请问这题怎么做?谢谢了……
在1/2处泰勒展开:
f(1) = f(1/2)+f’(1/2)*1/2+f’’(1/2)/2*(1/2)^2 +f’’’(t)/6*(1/2)^3
= f(1/2) + f’’(1/2)/8+f’’’(t)/48,
其中 1/2<t<1
类似,有:
f(0)= f(1/2) + f’’(1/2)/8-f’’’(s)/48,
其中 0<s<1/2
两式向减得:
2-1 = (f’’’(s)+f’’’(t))/48
f’’’(s)+f’’’(t)= 48
所以 2max{|f’’’(s)|,|f’’’(t)|}>=
|f’’’(s)|+|f’’’(t)|>=f’’’(s)+f’’’(t)= 48
==> max{|f’’’(s)|,|f’’’(t)|}>= 24
所以结论成立.
在1/2处泰勒展开:
f(1) = f(1/2)+f’(1/2)*1/2+f’’(1/2)/2*(1/2)^2 +f’’’(t)/6*(1/2)^3
= f(1/2) + f’’(1/2)/8+f’’’(t)/48,
其中 1/2<t<1
类似,有:
f(0)= f(1/2) + f’’(1/2)/8-f’’’(s)/48,
其中 0<s<1/2
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在1/2处泰勒展开:
f(1) = f(1/2)+f’(1/2)*1/2+f’’(1/2)/2*(1/2)^2 +f’’’(t)/6*(1/2)^3
= f(1/2) + f’’(1/2)/8+f’’’(t)/48,
其中 1/2<t<1
类似,有:
f(0)= f(1/2) + f’’(1/2)/8-f’’’(s)/48,
其中 0<s<1/2
两式向减得:
2-1 = (f’’’(s)+f’’’(t))/48
f’’’(s)+f’’’(t)= 48
所以 2max{|f’’’(s)|,|f’’’(t)|}>=
|f’’’(s)|+|f’’’(t)|>=f’’’(s)+f’’’(t)= 48
==> max{|f’’’(s)|,|f’’’(t)|}>= 24
所以结论成立。
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